New Insights into the Multifractal Formalism of Branching Random Walks on Galton–Watson Tree

Abstract

In the present work, we consider three branching random walk $S_n\mathsf{Z}\left(\mathsf{t}\right),$$\mathsf{Z}\in \{\mathsf{X},\mathsf{Y},\Phi \}$ on a supercritical random Galton–Watson tree $\partial \mathsf{T}$. We compute the Hausdorff and packing dimensions of the level set ${\mathrm{E}}_{\chi }(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}=\beta \right\},$ where $\partial \mathsf{T}$ is endowed with random metric using $S_n\Phi \left(\mathsf{t}\right)$. This is achieved by constructing a suitable Mandelbrot measure supported on $\mathrm{E}(\alpha ,\beta )$. In the case where $\Phi =1$, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting.

1. Introduction

The origins of multifractal analysis date back to the 1980s, with the pioneering work of B. Mandelbrot on multiplicative cascades in the study of energy dissipation in turbulence [1,2]. Since then, this theory has been extensively developed by numerous authors, establishing its central role in understanding the local geometric and scaling properties of measures. In particular, the multifractal analysis of a given measure $\mu$, eventually Borel and finite on a metric space $(\mathbb{X},d)$, passes through its local dimension or pointwise Hölder exponent. More precisely, for $\alpha \ge 0$, we define the level set

$${\mathrm{E}}_{\mu }\left(\alpha \right)=\left\{x\in \mathrm{supp}\mu :\underset{r\to 0}{\lim }{\displaystyle \frac{\log \mu \left(B\right(x,r\left)\right)}{\log r}}=\alpha \right\}$$

where $\mathrm{supp}\mu$ is the topological support of $\mu$, and $B(x,r)$ denotes the closed ball with center x and radius r. The level set ${\mathrm{E}}_{\mu }\left(\alpha \right)$ contains crucial information on the geometrical properties of $\mu$. The aim of multifractal analysis of a measure is to relate the Hausdorff and packing dimensions of these level sets, denoted, respectively, by $\dim \left({\mathrm{E}}_{\mu }\left(\alpha \right)\right)$ and $\mathrm{Dim}\left({\mathrm{E}}_{\mu }\left(\alpha \right)\right)$ to the Legendre transform of some convex function [3,4,5,6]. Moreover, obtaining a valid variant of the multifractal formalism does not require restricting to measures defined by simple power-law scalings in terms of the radius. In a broader framework, Cole [7] introduced the idea of controlling a measure $\mu$ by means of another suitable reference measure $\nu$, leading to a relative multifractal analysis based on the study of relative singularity sets. More specifically, he studied, for $\alpha \ge 0$, the size of the set

$${\mathrm{E}}_{\mu ,\nu }\left(\alpha \right)=\left\{x\in \mathrm{supp}\mu \cap \mathrm{supp}\nu :\underset{r\to 0}{\lim }{\displaystyle \frac{\log \mu \left(B\right(x,r\left)\right)}{\log \nu \left(B\right(x,r\left)\right)}}=\alpha \right\}.$$

(1)

This is achieved by introducing a generalized Hausdorff and packing measures denoted as ${\mathcal{H}}_{\mu ,\nu }^{q,s}$ and ${\mathcal{P}}_{\mu ,\nu }^{q,s}$, respectively (see [7,8] for more details on these measures). The multifractal formalism has been rigorously proven for many kinds of measures such as self-conformal measures, self-similar measures, self-affine measures, and for Moran measures [5,9,10,11,12,13,14,15,16,17,18]. The interesting authors may also consult the multifractal formalism of functions [19,20].

In this work, we will define the level sets using the branching random walks $S_n\mathsf{X}\left(\mathsf{t}\right)$, $S_n\mathsf{Y}\left(\mathsf{t}\right)$, and $S_n\Phi \left(\mathsf{t}\right)$ defined on the boundary of a Galton–Watson tree $\partial \mathsf{T}$ with defining number N (see definition in Section 2) and then generalizing the results given in [21,22,23,24]. More precisely, for $\alpha ,\beta \in \mathbb{R}$, we define the sets

$${\mathrm{E}}_{\chi }\left(\alpha \right)=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \right\}$$

(2)

and

$${\mathrm{E}}_{\chi }(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}=\beta \right\},$$

(3)

where $\chi =(\mathsf{X},\mathsf{Y})$. In [23], the set $\mathrm{E}\chi \left(\alpha \right)$ was studied by computing its Hausdorff and packing dimensions in the case $\mathsf{Y}=1$. This result was later refined in [21,24] by analyzing the speed of convergence of $S_n\mathsf{X}\left(\mathsf{t}\right)$ to $\alpha$. More precisely, the authors investigated the set of $\mathsf{t}\in \partial \mathsf{T}$ such that $S_n\mathsf{X}\left(\mathsf{t}\right)-n\alpha \sim s_n$ for a suitable sequence $\left(s_n\right)$. Subsequently, in [22], the author considered the sets $\mathrm{E}\chi \left(\alpha \right)$ and $\mathrm{E}\chi (\alpha ,\beta )$ when $\partial \mathsf{T}$ is endowed with the standard metric $d_1$ (10). In the present work, we extend these results to the metric distance $d_{\Phi }$ (11). In fact, one can prove that the set ${\mathrm{E}}_{\chi }\left({\alpha }_0\right)$ has full Hausdorff dimensions in $\partial \mathsf{T}$ for some ${\alpha }_0$, and it is worth investigating the existence of other branches over which ${\lim }_{n\to \infty }{S}_n\mathsf{X}\left(\mathsf{t}\right)/S_n\mathsf{Y}\left(\mathsf{t}\right)=\alpha \ne {\alpha }_0$. Moreover, studying the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$ provides connections between ergodic theory and geometric measure theory. In addition, our study can be extended to the case of vector-valued branching random walks, but we focus on the real case for simplicity.

Remark 1.

In [22], the author studied the set ${\mathrm{E}}_{\chi }(\alpha ,\beta )$ when $\partial \mathsf{T}$ is equipped with the standard metric $d_1$. This work is closely related to the analysis of ${\mathrm{E}}_{\chi }\left(\alpha \right)$ under the random metric $d_{\mathsf{Y}}$, which can be carried out under condition (4). In the present study, we consider a different situation in which the random metric is defined using a random variable distinct from both $\mathsf{X}$ and $\mathsf{Y}$. This requires an appropriate choice of parameters to construct the Mandelbrot measure supported on the set ${\mathrm{E}}_{\chi }(\alpha ,\beta )$.

Assume that $S_n\mathsf{Y}\left(\mathsf{t}\right)=n$, then the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$, denoted simply by ${\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)$, was studied in [23] under a standard metric $d_1$ defined by (10). The result is given through the convex function defined on $\mathbb{R}$ by

$$\tilde{P}\left(q\right)=\log \mathbb{E}\left(\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i\right)\right).$$

Assume that $\tilde{P}\left(q\right)<\infty$ over $\mathbb{R}$ and define the set $I_{\mathsf{X}}=\{\alpha \in \mathbb{R},{\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)\ne \varnothing \}$. Then, $I_{\mathsf{X}}$ is a non-empty convex compact set and can be described as $I_{\mathsf{X}}=\{\alpha \in \mathbb{R}:{\tilde{P}}^{\ast }\left(\alpha \right)\ge 0\},$ where ${\tilde{P}}^{\ast }\left(\alpha \right)$ is the Legendre transform of the function $\tilde{P}$ and, for $f:\mathbb{R}\to \mathbb{R}\cup \{\infty \}$ and $\alpha \in \mathbb{R}$, $f^{\ast }\left(\alpha \right)=\inf \{f\left(q\right)-q\alpha :q\in \mathbb{R}\}$ [21,22,24,25,26]. In addition, we almost surely (a.s.), for each $\alpha \in I_{\mathsf{X}}$, have that

$$\dim {\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)=\mathrm{Dim}{\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)={\tilde{P}}^{\ast }\left(\alpha \right).$$

Moreover, if $\alpha \in int\left(I_{\mathsf{X}}\right)$, the interior of the set $I_{\mathsf{X}}$, then ${\tilde{P}}^{\ast }\left(\alpha \right)>0$. Assume that there exists $\gamma >0$ such that

$$\mathbb{E}\left(\sum _{i=1}^N\exp (-\gamma {\mathsf{Z}}_i)\right)<1\mathrm{for}\mathrm{all}\mathsf{Z}\in \{\Phi ,\mathsf{Y}\},$$

(4)

which holds, for example, by dominated convergence: since ${\sum }_{i=1}^N{e}^{-\gamma {\mathsf{Z}}_i}$ decreases pointwise to 0 as $\gamma \to \infty$ and is dominated by the integrable function ${\sum }_{i=1}^N{e}^{-{\gamma }_0{\mathsf{Z}}_i}$ for some ${\gamma }_0>0$, we have

$$\mathbb{E}\left(\sum _{i=1}^N{e}^{-\gamma {\mathsf{Z}}_i}\right)⟶0,$$

so the expectation is strictly less than 1 for large enough $\gamma$. Then, one may define the random ultrametric distance $d_{\Phi }$ on $\partial \mathsf{T}$, and $(\partial \mathsf{T},d_{\varphi })$ is compact (see definition (11)). Such metrics are used to obtain the geometric realization of Mandelbrot measures on random self-similar sets satisfying some separation conditions [9,27]. In addition (4) is also necessary to construct the multifractal Hausdorff and packing measures in Section 2.1. The set ${\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)$ is studied in [25] when $\partial \mathsf{T}$ is endowed with random metric $d_{\Phi }$, defined by (11). Now, consider the case $S_n\mathsf{Y}\left(\mathsf{t}\right)=S_n\Phi \left(\mathsf{t}\right)$, then the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$ was considered in [22] under metric $d_{\Phi }$. This result can be seen as a counterpart of Cole’s result [7] when studying the sets ${\mathrm{E}}_{\mu ,\nu }\left(\alpha \right)$ defined by (1) in the case of the multifractal analysis of measures. In fact, it is difficult to compute the Hausdorff and packing dimensions of the set ${\mathrm{E}}_{\mu ,\nu }\left(\alpha \right)$ but only the Hausdorff and packing dimension with respect to the measure $\nu$. To solve this problem, Cole introduced and studied the sets

$${\mathrm{E}}_{\mu ,\nu }(\alpha ,\beta )=\left\{x\in \mathrm{supp}\mu \cap \mathrm{supp}\nu :\underset{r\to 0}{\lim }{\displaystyle \frac{\log \mu \left(B\right(x,r\left)\right)}{\log \nu \left(B\right(x,r\left)\right)}}=\alpha ;{\displaystyle \frac{\log \nu \left(B\right(x,r\left)\right)}{\log r}}=\beta \right\},$$

(5)

for all $\alpha ,\beta \ge 0$. Notice that, under (4), (see Section 4) that

$${\mathbb{E}}_{\chi }\left(\alpha \right)=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\left(\mathsf{X}\left(t\right)-\alpha \mathsf{Y}\left(t\right)\right)}{n}}=0\right\}$$

and then (recall set ${\mathrm{E}}_{\mathsf{X}}\left(\alpha \right)$), $\alpha \in I_{\chi }≔\left\{\alpha ;{\mathbb{E}}_{\chi }\left(\alpha \right)\ne \varnothing \right\}$ if ${\inf }_{q\in \mathbb{R}}\Pi (q,\alpha )\ge 0$, where

$$\Pi (q,\alpha )≔\log \left(\mathbb{E}\sum _{i=1}^N{e}^{q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i}\right).$$

(6)

This will be an important tool for computing the Hausdorff and packing dimension of ${\mathrm{E}}_{\chi }\left(\alpha \right)$ and then deducing the dimensions of the set ${\mathrm{E}}_{\chi }(\alpha ,\beta )$. To this end, we will construct an adequate Mandelbrot measure supported by the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$ [22,28] ( the reader can consult [5,6,13,14,29] for the construction of auxiliary measures in different contexts). We define, for all $(q,\alpha ,t)\in {\mathbb{R}}^3$,

$$S_{\alpha }(q,t)=\sum _{i=1}^N\exp \left(q({\mathsf{X}}_i-\alpha {\mathsf{Y}}_i)-t{\Phi }_i\right).$$

We define $\alpha \in int\left(I_{\mathsf{X}}\right)$, ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right))$ as satisfiying $S_{\alpha }(q,{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)))=1$. We assume that

$$\forall (q,\alpha )\in {\mathbb{R}}^2,\mathrm{such}\mathrm{that}{\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)\right)>0,\exists \gamma >1,\mathbb{E}\left(S_{\alpha }{(q,{\tilde{\mathcal{P}}}_{\alpha }\left(q\right))}^{\gamma }\right)<\infty .$$

(7)

and

$$\underset{q\in \mathbb{R}}{\sup }\mathbb{E}\left(\sum _{i=1}^N{\mathsf{Y}}_i\exp \left(q{\mathsf{X}}_i\right)\right)<\infty .$$

(8)

which will be used to estimate the dimension of the Mandelbrot measure from below (Theorem 1).

This work extends the multifractal analysis of branching random walks on Galton-Watson trees to a framework involving random metrics. By constructing suitable Mandelbrot measures, we provide a unified approach for studying the Hausdorff and packing dimensions of the level sets ${\mathrm{E}}_{\chi }\left(\alpha \right)$ and ${\mathrm{E}}_{\chi }(\alpha ,\beta )$, generalizing previous results and connecting ergodic theory with geometric measure theory. In Section, Section 3, under the metric $d_1$, we give a new formalism in this setting that is parallel to that of vector functions [6,19,26,30,31] or measures [5,7,22]. In Section 4, we study these sets when $\partial \mathsf{T}$ is endowed with the distance $d_{\Phi }$. This generalizes the results proven in [22,23,24].

2. Notations and Preliminary Results

Let ${\mathbb{N}}_{+}$ be the set of positive integers and $(\Omega ,\mathcal{A},\mathbb{P})$ be a probability space. We consider the following random vector $(N,({\mathsf{X}}_1,{\mathsf{Y}}_1,{\Phi }_1),({\mathsf{X}}_2,{\mathsf{Y}}_2,{\Phi }_2),\dots )$ with independent components, taking values in $\mathbb{N}\times {(\mathbb{R}\times {\mathbb{R}}_{+}^{\ast }\times {\mathbb{R}}_{+}^{\ast })}^{{\mathbb{N}}_{+}}$. Consider a family of independent copies of this random vector

$${\left\{(N_{u0},({\mathsf{X}}_{u1},{\mathsf{Y}}_{u1},{\Phi }_{u1}),({\mathsf{X}}_{u2},{\mathsf{Y}}_{u2},{\Phi }_{u2}),\dots )\right\}}_{u\in {\bigcup }_{n\ge 0}{\mathbb{N}}_{+}^n}$$

indexed by the finite sequences $u=u_1\dots u_n,n\ge 0,u_i\in {\mathbb{N}}_{+}^{\ast }$ ($n=0$ corresponds to the empty sequence denoted ∅). In our study, we avoid the trivial case and assume that

$$\nexists c\in \mathbb{R},{\mathsf{X}}_i=c{\mathsf{Y}}_i\forall 1\le i\le N\mathrm{a}.\mathrm{s}.;$$

(9)

otherwise, our study is reduced to a one-parameter case when we study $E\left(\alpha \right)$. Let $\mathsf{T}$ be the Galton–Watson tree with defining element $\left\{N_u\right\}$: we have $\varnothing \in \mathsf{T}$ and, if $u\in \mathsf{T}$ and $i\in {\mathbb{N}}_{+}$, then $ui$, the concatenation of u and i, belongs to $\mathsf{T}$ if and only if $1\le i\le N_u$. Similarly, for each $u\in {\cup }_{n\ge 0}{\mathbb{N}}_{+}^n$, we denote by $\mathsf{T}\left(u\right)$ the Galton–Watson tree rooted at u defined by $\left\{N_{uv}\right\},v\in {\cup }_{n\ge 0}{\mathbb{N}}_{+}^n.$ For each $u\in {\bigcup }_{n\ge 0}{\mathbb{N}}_{+}^n$, let $\left|u\right|$ denote its length, i.e., the number of letters of u, and $\left[u\right]$ denote the cylinder $u·{\mathbb{N}}_{+}^{{\mathbb{N}}_{+}},$ i.e., the set of $\mathsf{t}\in {\mathbb{N}}_{+}^{{\mathbb{N}}_{+}}$ such that ${\mathsf{t}}_1{\mathsf{t}}_2\dots {\mathsf{t}}_{\left|u\right|}=u$. If $\mathsf{t}\in {\mathbb{N}}_{+}^{{\mathbb{N}}_{+}}$, we set $\left|\mathsf{t}\right|=\infty$, and the set of prefixes of $\mathsf{t}$ consists of $\{\varnothing \}\cup \{{\mathsf{t}}_1{\mathsf{t}}_2\dots {\mathsf{t}}_n:n\ge 1\}\cup \left\{t\right\}.$ Also, we set ${\mathsf{t}}_{|n}={\mathsf{t}}_1\dots {\mathsf{t}}_n$ if $n\ge 1$ and ${\mathsf{t}}_{|0}=\varnothing$.

We will assume that $\mathbb{E}\left(N\right)>1$ and $\mathbb{P}(N\ge 1)=1$ so that the Galton–Watson tree is supercritical with a probability of extinction equaling 0, where $\mathbb{E}$ denotes the expectation with respect to $\mathbb{P}$. The boundary of $\mathsf{T}$ is the subset of ${\mathbb{N}}_{+}^{{\mathbb{N}}_{+}}$ defined as

$$\partial \mathsf{T}=\bigcap _{n\ge 1}\bigcup _{u\in {\mathsf{T}}_n}\left[u\right],$$

where ${\mathsf{T}}_n=\mathsf{T}\cap {\mathbb{N}}_{+}^n$. The set ${\mathbb{N}}_{+}^{{\mathbb{N}}_{+}}$ is endowed with the standard ultrametric distance

$$d_1:(s,\mathsf{t})⟼\exp (-|s\wedge \mathsf{t}\left|\right),$$

(10)

where $s\wedge t$ stands for the longest common prefix of s and t, with the convention that $\exp (-\infty )=0$. Endowed with the induced distance, the set $\partial \mathsf{T}$ is a.s. compact. For $\mathsf{t}\in \partial \mathsf{T}$, we set the branching random walks

$${\displaystyle S_n\mathsf{X}\left(\mathsf{t}\right)=\sum _{k=1}^n{\mathsf{X}}_{{\mathsf{t}}_1\cdots {\mathsf{t}}_k},S_n\Phi \left(\mathsf{t}\right)=\sum _{k=1}^n{\Phi }_{{\mathsf{t}}_1\cdots {\mathsf{t}}_k}{S}_n\mathsf{Y}\left(\mathsf{t}\right)=\sum _{k=1}^n{\mathsf{Y}}_{{\mathsf{t}}_1\cdots {\mathsf{t}}_k}.}$$

Since the branching random walk $S_n\mathsf{Z}\left(\mathsf{t}\right)$ for all $\mathsf{Z}\in \{\mathsf{X},\mathsf{Y},\Phi \}$ depends on ${\mathsf{t}}_1\cdots {\mathsf{t}}_n$ only, we also denote by $S_n\mathsf{Z}\left(u\right)$ the constant value of $S_n\mathsf{Z}(·)$ over $\left[u\right]$ whenever $u\in {\mathsf{T}}_n$. In addition, for each $\mathsf{t}\in \partial \mathsf{T}$ and $r>0$, we have the closed ball $B(t,r)=[{\mathsf{t}}_1,{\mathsf{t}}_2,\dots ,{\mathsf{t}}_n]$, where n is the unique integer satisfying

$$e^{-n}\le r<e^{-(n-1)}.$$

Assume (4), then there exista $0<C_1<C_2<1$ such that, with a probability of 1, for large enough n, we have ([25], Lemma 2.1)

$$C_1^n\le \min \{\exp (-S_n\Phi \left(u\right)):u\in {\mathsf{T}}_n\}\le \max \{\exp (-S_n\Phi \left(u\right)):u\in {\mathsf{T}}_n\}\le C_2^n.$$

It follows that, with a probability of 1, $S_n\Phi \left(u\right)$ tends to ∞ uniformly in $u\in {\mathsf{T}}_n$ as $n\to \infty$, so that we obtain the random ultrametric distance

$$d_{\Phi }:(s,\mathsf{t})⟼\exp (-S_{|s\wedge t|}\Phi (s\wedge \mathsf{t})),$$

(11)

on $\partial \mathsf{T}$, and $(\partial \mathsf{T},d_{\varphi })$ is compact.

2.1. Generalized Hausdorff and Packing Measures and Dimensions

In the following, we will set up, for $q,t\in \mathbb{R}$, the multifractal Hausdorff and packing measures and dimensions. Let $A\subseteq \partial \mathsf{T}$ and $\chi =(\mathsf{X},\mathsf{Y})$. For $r>0$, we define the function

$${\mathcal{H}}_{\chi ,n}^{q,t}\left(A\right)=\inf \left\{\sum _{i\in \mathcal{I}}{e}^{q{S}_{|v_i|}\mathsf{X}\left(v_i\right)-t{S}_{|v_i|}\mathsf{Y}\left(v_i\right)}\right\}$$

$${\mathcal{H}}_{\chi }^{q,t}\left(A\right)=\underset{n\to \infty }{\lim }{\mathcal{H}}_{\chi ,n}^{q,t}\left(A\right),$$

where ${\left\{\left[v_i\right]\right\}}_{i\in \mathcal{I}}$ is a $e^{-n}$-covering of A. We take ${\mathcal{H}}_{\chi }^{q,t}(\varnothing )=0$ and we define the Hausdorff dimension

$${\dim }_{\chi }^q\left(A\right)≔\inf \left\{t\in \mathbb{R};{\mathcal{H}}_{\chi }^{q,t}\left(A\right)=0\right\}.$$

We denote ${\mathsf{b}}_{\chi }\left(q\right)≔{\dim }_{\chi }^q(\partial \mathsf{T}).$ In particular, if $q=0$ and $t>0$, then ${\mathcal{H}}_{\chi }^{q,t}$ is denoted by ${\mathcal{H}}_{\mathsf{Y}}^t$, and ${\dim }_{\chi }^q$ is denoted by ${\dim }_{\mathsf{Y}}$. In addition, if $\mathsf{Y}=1$, then ${\mathcal{H}}_{\mathsf{Y}}^t$ is the standard Hausdorff measure (under the standard metric $d_1$), which is denoted by ${\mathcal{H}}^t$, and ${\dim }_{\mathsf{Y}}$ is denoted by $\dim .$ Similarly, we define

$${\mathcal{P}}_{\chi ,n}^{q,t}\left(A\right)=\sup \left\{\sum _i{e}^{q{S}_{|v_i|}\mathsf{X}\left(x_i\right)-t{S}_{|v_i|}\mathsf{Y}\left(x_i\right)}\right\}$$

$${\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A\right)=\underset{r\to 0}{\lim }{\mathcal{P}}_{\chi ,n}^{q,t}\left(A\right),$$

where ${\left\{\left[v_i\right]\right\}}_{i\in \mathcal{I}}$ is an $e^{-n}$-packing of A, and we take ${\overline{\mathcal{P}}}_{\chi }^{q,t}(\varnothing )=0$. The function ${\overline{\mathcal{P}}}_{\chi }^{q,t}$ is not $\sigma$ additive; for this, we define the ${\mathcal{P}}_{\chi }^{q,t}$ measure as

$${\mathcal{P}}_{\chi }^{q,t}\left(A\right)=\inf \left\{\sum _i{\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A_i\right)|A\subseteq \bigcup _i{A}_i\right\}.$$

We may define the multifractal packing dimension by

$${\mathrm{Dim}}_{\chi }^q\left(A\right)≔\inf \left\{t\in \mathbb{R};{\mathcal{P}}_{\chi }^{q,t}\left(A\right)=0\right\}.$$

We will denote ${\mathsf{B}}_{\chi }\left(q\right)≔{\mathrm{Dim}}_{\chi }^q(\partial \mathsf{T}).$ In particular, if $q=0$ and $t>0$, then ${\mathcal{P}}_{\chi }^{q,t}$ will be denoted by ${\mathcal{P}}_{\mathsf{Y}}^t$, and ${\mathrm{Dim}}_{\chi }^q$ will be denoted by ${\mathrm{Dim}}_{\mathsf{Y}}$. In addition, if $\mathsf{Y}=1$, then ${\mathcal{P}}_{\mathsf{Y}}^t$ is the standard packing measure, which will be denoted by ${\mathcal{P}}^t$, and ${\mathrm{Dim}}_{\mathsf{Y}}$ will be denoted by $\mathrm{Dim}$.

Definition 1.

Define ${\Delta }_{\chi }^q\left(A\right)≔\inf \left\{t\in \mathbb{R};{\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A\right)=0\right\}$ and ${\Lambda }_{\chi }\left(q\right)={\Delta }_{\chi }^q(\partial \mathsf{T})$,

Remark 2.

• Recall (11), and since we assume (4), then one can define the metric $d_{\mathsf{Y}}$ on $\partial \mathsf{T}$. Assume that $\partial \mathsf{T}$ is endowed with the metric $d_{\mathsf{Y}}$, then the standard Hausdorff and packing measures are ${\mathcal{H}}_{\mathsf{Y}}^t$ and ${\mathcal{P}}_{\mathsf{Y}}^t$.
• We have

  $${\dim }_{\chi }^q\left(A\right)\le {\mathrm{Dim}}_{\chi }^q\left(A\right)\le {\Delta }_{\chi }^q\left(A\right)\mathit{so}\mathit{that}{\mathsf{b}}_{\chi }\left(q\right)\le {\mathsf{B}}_{\chi }\left(q\right)\le {\Lambda }_{\chi }\left(q\right).$$

Under hypothesis (4), we have, with a probability of 1, that $S_n\mathsf{Y}\left(u\right)$ tends to ∞ uniformly in $u\in {\mathsf{T}}_n$ as $n\to \infty$. Therefore, the functions ${\mathcal{H}}_{\chi }^{q,t}$ and ${\mathcal{P}}_{\chi }^{q,t}$ are special cases of those studied in [19]. It follows that the functions ${\mathcal{H}}_{\chi }^{q,t}$ and ${\mathcal{P}}_{\chi }^{q,t}$ are metric outer measures [19,26,30]. Moreover, there is a fascinating relationship between the multifractal measures and the standard measures when working on set ${\mathrm{E}}_{\chi }(\alpha ,\beta )$. This makes them very suitable measures for multifractal analysis. More precisely, we have the following lemma:

Lemma 1.

Let $\alpha ,q\in \mathbb{R}$ and $\beta \ge 0$.

• If ${\mathsf{B}}_{\chi }\left(q\right)-q\alpha \ge 0$, then $\mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta \left({\mathsf{B}}_{\chi }\left(q\right)-q\alpha \right).$
• If ${\mathsf{b}}_{\chi }\left(q\right)-q\alpha \ge 0$, then $\dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta \left({\mathsf{b}}_{\chi }\left(q\right)-q\alpha \right).$
• ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(\partial \mathsf{T}\setminus {\mathrm{E}}_{\chi }\left({\Lambda }_{\chi }^{\prime }\left(q\right)\right)\right)=0$, if ${\Lambda }_{\chi }^{\prime }\left(q\right)$ exists.

Proof. 

Let $\alpha ,t\in \mathbb{R}$, and $\beta \ge 0$, and let ${\epsilon }_1$ and ${\epsilon }_2$ be two positive numbers such that

$$\begin{array}{c}\hfill 0<t-q(\alpha +{\epsilon }_1)\mathrm{and}{\epsilon }_2\le \beta \left(t-q(\alpha -{\epsilon }_1)\right)\end{array}$$

(12)

For $m\ge 1$, we consider the set

$${\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)=\left\{\mathsf{t}\in \partial \mathsf{T};|{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}-\alpha |\le {\epsilon }_1\mathrm{and}|{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}-\beta |\le {\displaystyle \frac{{\epsilon }_2}{t-q(\alpha -\pm {\epsilon }_1)}}\mathrm{for}n\ge m\right\},$$

where ${\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)={\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)$ if $q\ge 0$ and ${\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)={\mathsf{A}}_m^{-}({\epsilon }_1,{\epsilon }_2)$ if $q<0$. Then, we have

$${\displaystyle {\mathrm{E}}_{\chi }\left(\alpha ,\beta \right)\subseteq \bigcap _{p_1,p_2\ge 1}\bigcup _{m\ge 1}{\mathsf{A}}_m^{\pm }(1/p_1,1/p_2).}$$

Now, it is enough to prove the following inequality:

$${\Pi }^{\beta \left(t-q(\alpha +{\epsilon }_1)\right)+{\epsilon }_2}\left({\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)\right)\le {\Pi }_{\chi }^{q,t}\left({\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)\right).$$

(13)

for all $\Pi \in \{\mathcal{H},\mathcal{P}\}$. Indeed, take, for example, $\Pi =\mathcal{H}$ and $t={\mathsf{b}}_{\chi }\left(q\right)+ϵ$ for $ϵ>0$. Then, using (13), we get

$${\Pi }_{\chi }^{q,t}\left({\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)\right)=0$$

and then $\dim \left({\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)\right)\le \beta \left(t-q(\alpha +{\epsilon }_1)\right)+{\epsilon }_2$. Then, the result follows by the countable stability of the Hausdorff (and packing is the second case) dimension. In the following we will prove (13). First, notice that if $\mathsf{t}\in {\mathsf{A}}_m^{\pm }({\epsilon }_1,{\epsilon }_2)\cap \left[u_i\right]$, then

$$\left\{\begin{array}{cc}{e}^{-|u_i|\left(\beta \left(t-q(\alpha -{\epsilon }_1)\right)+{\epsilon }_2\right)}\hfill & \le e^{-\left(t-q(\alpha -{\epsilon }_1)\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}(q\ge 0)\hfill \\ e^{-|u_i|\left(\beta \left(t-q(\alpha +{\epsilon }_1)\right)+{\epsilon }_2\right)}\hfill & \le e^{-\left(t-q(\alpha +{\epsilon }_1)\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}(q<0)\hfill \end{array}\right.$$

(14)

Moreover, we have

$$\left\{\begin{array}{cc}{e}^{q\left(\alpha -{\epsilon }_1\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}\hfill & \le e^{q{S}_{|u_i|}\mathsf{X}\left(u_i\right)}(q\ge 0)\hfill \\ e^{q\left(\alpha +{\epsilon }_1\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}\hfill & \le e^{q{S}_{|u_i|}\mathsf{X}\left(u_i\right)}(q\le 0)\hfill \end{array}\right.$$

(15)

(1)

Let l be a positive integer and ${\left(\left[u_i\right]\right)}_{i\in \mathcal{I}}$ be an $e^{-l}$ packing of ${\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)$. Using (15), for all $q\ge 0$, one has

$$\begin{array}{ccc}\hfill \sum _i{e}^{-|u_i|\left(\beta \left(t-q(\alpha -{\epsilon }_1)\right)+{\epsilon }_2\right)}& \le & \sum _i{e}^{-\left(t-q(\alpha -{\epsilon }_1)\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}\hfill \\ & \le & \sum _i{e}^{q{S}_{|u_i|}\mathsf{X}\left(u_i\right)-t{S}_{|u_i|}\mathsf{Y}\left(u_i\right)}.\hfill \end{array}$$

This clearly implies that ${\mathcal{P}}_l^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{P}}_{\chi ,l}^{q,t}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right).$ Letting $l\to \infty$ gives ${\overline{\mathcal{P}}}^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)\le {\overline{\mathcal{P}}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right).$ Finally, let ${\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)$${\displaystyle \subseteq \bigcup _i}$$A_i$. As we have

$$\begin{array}{ccc}\hfill {\mathcal{P}}^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)& \le & \sum _i{\overline{\mathcal{P}}}^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\cap A_i\right)\hfill \\ & \le & \sum _i{\overline{\mathcal{P}}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\cap A_i\right)\le \sum _i{\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A_i\right),\hfill \end{array}$$

then we obtain ${\mathcal{P}}^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{P}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)$. Similarly, we have

$${\mathcal{P}}^{\beta (t-q(\alpha +{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{-}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{P}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{-}({\epsilon }_1,{\epsilon }_2)\right)$$

for $q<0$.

(2)

Let $q\ge 0$. Let l be a positive integer and ${\left(\left[u_i\right]\right)}_{i\in \mathcal{I}}$ be an $e^{-l}$ covering of ${\mathsf{A}}_m({\epsilon }_1,{\epsilon }_2)$, with $0<e^{-l}\le {\displaystyle \frac{1}{m}}$. Using (15), for all $q\ge 0$, one has

$$\begin{array}{ccc}\hfill {\mathcal{H}}_l^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)& \le & \sum _i{e}^{-\left(t-q(\alpha -{\epsilon }_1)\right)S_{|u_i|}\mathsf{Y}\left(u_i\right)}\hfill \\ & \le & {\displaystyle \sum _i{e}^{q{S}_{|u_i|}\mathsf{X}\left(u_i\right)-t{S}_{|u_i|}\mathsf{Y}\left(u_i\right)}.}\hfill \end{array}$$

To this end, we have ${\mathcal{H}}_l^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{H}}_{\chi ,l}^{q,t}\left({\mathsf{A}}_m({\epsilon }_1,{\epsilon }_2)\right)$, and then ${\mathcal{H}}^{\beta (t-q(\alpha -{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{H}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{+}({\epsilon }_1,{\epsilon }_2)\right).$ Now if $q<0$, similarly, one can prove that

$${\mathcal{H}}^{\beta (t-q(\alpha +{\epsilon }_1))+{\epsilon }_2}\left({\mathsf{A}}_m^{-}({\epsilon }_1,{\epsilon }_2)\right)\le {\mathcal{H}}_{\chi }^{q,t}\left({\mathsf{A}}_m^{-}({\epsilon }_1,{\epsilon }_2)\right).$$

(3)

We consider sets A and B defined as

$$A=\left\{\mathsf{t}\in \partial \mathsf{T}|\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}>{\Lambda }_{\chi }^{\prime }\left(q\right)\right\}\mathrm{and}B=\left\{\mathsf{t}\in \partial \mathsf{T}|\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}<{\Lambda }_{\chi }^{\prime }\left(q\right)\right\}.$$

We will only have to prove that

$${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(A\right)={\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(B\right)=0.$$

(16)

In what follows we will prove (16) for set A, and it will be similar for B. Let $\lambda ={\Lambda }^{\prime }\left(q\right)+\eta$, then

$$A\subseteq \bigcup _{\eta >0}\bigcup _{m\ge 1}{A}_m≔\bigcup _{\eta >0}\bigcup _{m\ge 1}\left\{\mathsf{t}\in \partial \mathsf{T}|{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}>\lambda ;n\ge m\right\}.$$

Since $\lambda >{\Lambda }_{\chi }^{\prime }\left(q\right)$, choose $t>0$ such that ${\Lambda }_{\chi }(q+t)<{\Lambda }_{\chi }\left(q\right)+t\lambda$. It follows, by definition of ${\Lambda }_{\chi }(q+t)$, that

$${\overline{\mathcal{P}}}_{\chi }^{q+t,{\Lambda }_{\chi }\left(q\right)+t\lambda }(\partial \mathsf{T})=0.$$

Let $\left\{\left[u_j\right]\right\}$ be an $e^{-l}$ covering of $A_m$ for $m\ge 1$. Using Besicovitch’s covering theorem, we can construct $\zeta$ finite sub-families such that

$$A_m\subseteq \bigcup _{i=1}^{\zeta }\bigcup _j\left[u_{ij}\right]\mathrm{and}{\left\{\left[u_j\right]\right\}}_j\mathrm{is}\mathrm{an}{e}^{-l}\text{-}\mathrm{packing}\mathrm{of}{A}_m.$$

It follows that

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{\chi ,l}^{q,{\Lambda }_{\chi }\left(q\right)}\left(A_m\right)& \le & {\displaystyle \sum _{1\le i\le \zeta }\sum _j{e}^{\left(q{S}_{\left|u_{ij}\right|}\mathsf{X}\left(u_{ij}\right))-{\Lambda }_{\chi }\left(q\right)S_{|u_{ij}|}\mathsf{Y}\left(u_{ij}\right)\right)}}\hfill \\ & \le & {\displaystyle \sum _{1\le i\le \zeta }\sum _j{e}^{\left((q+t)S_{|u_{ij}|}\mathsf{X}\left(u_{ij}\right)-({\Lambda }_{\chi }\left(q\right)+\lambda t)S_{|u_{ij}|}\mathsf{Y}\left(u_{ij}\right)\right)}}\hfill \\ & \le & \zeta {\overline{\mathcal{P}}}_{\chi ,l}^{q+t,{\Lambda }_{\chi }\left(q\right)+\lambda t}\left(A_m\right),\hfill \end{array}$$

which implies that ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(A_m\right)\le \zeta {\overline{\mathcal{P}}}_{\chi }^{q+t,{\Lambda }_{\chi }\left(q\right)+\lambda t}\left(A_m\right)=0.$ Similarly, for $\lambda ={\Lambda }^{\prime }\left(q\right)-\eta$, then

$$B\subseteq \bigcup _{\eta >0}\bigcup _{m\ge 1}{B}_m≔\bigcup _{\eta >0}\bigcup _{m\ge 1}\left\{\mathsf{t}\in \partial \mathsf{T}|{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}<\lambda ;n\ge m\right\}.$$

Since $\lambda <{\Lambda }_{\chi }^{\prime }\left(q\right)$, choose $t>0$ such that ${\Lambda }_{\chi }(q-t)<{\Lambda }_{\chi }\left(q\right)-t\lambda$. It follows, by definition of ${\Lambda }_{\chi }(q+t)$, that

$${\overline{\mathcal{P}}}_{\chi }^{q-t,{\Lambda }_{\chi }\left(q\right)-t\lambda }(\partial \mathsf{T})=0.$$

Let $\left\{\left[u_j\right]\right\}$ be an $e^{-l}$ covering of $B_m$ for $m\ge 1$. Using Besicovitch’s covering theorem, we can construct $\zeta$ finite sub-families such that,

$$B_m\subseteq \bigcup _{i=1}^{\zeta }\bigcup _j\left[u_{ij}\right]\mathrm{and}{\left[u_j\right]}_j\mathrm{is}\mathrm{an}{e}^{-l}\mathrm{packing}\mathrm{of}{B}_m.$$

It follows that,

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{\chi ,l}^{q,{\Lambda }_{\chi }\left(q\right)}\left(B_m\right)& \le & {\displaystyle \sum _{1\le i\le \zeta }\sum _j{e}^{\left(q{S}_{|u_{ij}|}\mathsf{X}\left(u_{ij}\right))-{\Lambda }_{\chi }\left(q\right)S_{|u_{ij}|}\mathsf{Y}\left(u_{ij}\right)\right)}}\hfill \\ & \le & {\displaystyle \sum _{1\le i\le \zeta }\sum _j{e}^{\left((q-t)S_{|u_{ij}|}\mathsf{X}\left(u_{ij}\right)-({\Lambda }_{\chi }\left(q\right)-\lambda t)S_{|u_{ij}|}\mathsf{Y}\left(u_{ij}\right)\right)}}\hfill \\ & \le & \zeta {\overline{\mathcal{P}}}_{\chi ,l}^{q-t,{\Lambda }_{\chi }\left(q\right)-\lambda t}\left(B_m\right).\hfill \end{array}$$

This implies that ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(B_m\right)\le \zeta {\overline{\mathcal{P}}}_{\chi }^{q-t,{\Lambda }_{\chi }\left(q\right)-\lambda t}\left(B_m\right)=0.$

□

We define

$${\mathcal{P}}_{\chi ,n}^{\ast q,t}\left(A\right)=\sup \left\{\sum _i{e}^{q{S}_{|v_i|}\mathsf{X}\left(x_i\right)-t{S}_{|v_i|}\mathsf{Y}\left(x_i\right)}\right\}$$

$${\mathcal{P}}_{\chi }^{\ast q,t}\left(A\right)=\underset{n\to \infty }{\lim }{\mathcal{P}}_{\chi ,n}^{\ast q,t}\left(A\right),$$

where ${\left\{\left[v_i\right]\right\}}_{i\in \mathcal{I}}$ is a packing of A with $|\left[v_i\right]|=e^{-n}$

$${\Delta }_{\chi }^{\ast q}\left(A\right)=\inf \left\{t\ge 0|{\mathcal{P}}_{\chi }^{\ast q,t}\left(A\right)=0\right\}.$$

Lemma 2.

Let $q,t\in \mathbb{R}$ and $n\ge 1$. Then, we have

• ${\displaystyle {\mathcal{P}}_{\chi ,n}^{\ast q,t}(\partial \mathsf{T})=\sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right)}.}$
• ${\Delta }_{\chi }^q\left(A\right)={\Delta }_{\chi }^{\ast q}\left(A\right)$ for all  $A\in \partial \mathsf{T}.$
• ${\displaystyle {\Lambda }_{\chi }\left(q\right)=\inf \left\{t\in \mathbb{R},\underset{n\to \infty }{lim\; sup}\frac{1}{n}\log \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right)}\le 0\right\}.}$

Proof. 

1.

Let ${\left\{\left[u_j\right]\right\}}_j$ be a packing of $\partial \mathsf{T}$ such that $|\left[u_j\right]|=e^{-n}$, then

$$\begin{array}{ccc}\hfill {\displaystyle \sum _j{e}^{q{S}_n\mathsf{X}\left(u_j\right))-t{S}_n\mathsf{Y}\left(u_j\right)}}& \le & {\displaystyle \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right)}.}\hfill \end{array}$$

It follows that ${\displaystyle {\mathcal{P}}_{\chi ,n}^{\ast q,t}(\partial \mathsf{T})\le \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right)}.}$ On the other hand, since $\left\{\left[u\right],u\in {\mathsf{T}}_n\right\}$ is an $e^{-n}$ packing of $\partial \mathsf{T}$, we have

$${\displaystyle \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(\left[u\right]\right)-t{S}_n\mathsf{Y}\left(\left[u\right]\right)}\le {\mathcal{P}}_{\chi ,n}^{\ast q,t}(\partial \mathsf{T}),}$$

as required.

2.

Since ${\mathcal{P}}_{\chi }^{\ast q,t}\left(A\right)\le {\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A\right)$, one has ${\Delta }_{\chi }^{\ast q}\left(A\right)\le {\Delta }_{\chi }^q\left(A\right)$. Now, suppose that ${\Delta }_{\chi }^q\left(A\right)>0$; otherwise, we are finished. Let t and $ϵ$ be two positive numbers such that $0<t-ϵ<t<{\Delta }_{\chi }^q\left(A\right)$. Therefore, ${\overline{\mathcal{P}}}_{\chi }^{q,t}\left(A\right)=+\infty$. Then, there exists $n_0$ such that, for all $n>n_0$, there exists an $e^{-n}$ packing of A with

$$\sum _j{e}^{q{S}_{|u_j|}\mathsf{X}\left(u_j\right)-t{S}_{|u_j|}\mathsf{Y}\left(u_j\right)}\ge 1.$$

As

$$\sum _j{e}^{q{S}_{|u_j|}\mathsf{X}\left(u_j\right)-(t-ϵ)S_{|u_j|}\mathsf{Y}\left(u_j\right)}=\sum _{j\ge 0}\sum _{|u_j|=e^{-n-j}}{e}^{q{S}_{|u_j|}\mathsf{X}\left(u_j\right)-(t-ϵ)S_{|u_j|}\mathsf{Y}\left(u_j\right)}.$$

Then, there exists $i\ge 0$ such that

$$\sum _{|u_j|=e^{-n-j}}{e}^{q{S}_{|u_j|}\mathsf{X}\left(u_j\right)-(t-ϵ)S_{|u_j|}\mathsf{Y}\left(u_j\right)}\ge e^{-jϵ}(1-e^{-ϵ}).$$

(17)

Notice, since we assume (4), then there exists $0<C_1<C_2<1$ such that, with a probability of 1 for large enough n, we have ([25], Lemma 2.1)

$$C_1^n\le \min \left\{e^{-S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\right\}\le \max \left\{e^{-S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\right\}\le C_2^n.$$

(18)

Indeed, let $r_n=\max \{e^{-S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\}$ and $r_n^{\prime }=\min \{e^{-S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\}$ for $n\ge 1$. For each $\beta \in {\mathbb{R}}_{+}^{\ast }$ and $\gamma >0$, we have

$$\begin{array}{ccc}\hfill \mathbb{P}(r_n\ge {\beta }^n)& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{-S_n\mathsf{Y}\left(u\right)}{\beta }^{-n}\ge 1\}\right)\le {\beta }^{-n\gamma }{\left(\mathbb{E}\left(S(0,\gamma )\right)\right)}^n,\hfill \end{array}$$

where $S(0,\gamma )$ is defined in Section 3. Since ${\mathsf{Y}}_i$ are positives and (4) is assumed to be realized, then there exists ${\gamma }_0>0$ such that $\mathbb{E}\left(S(0,{\gamma }_0)\right)<1$. Consequently, if $\beta \in (\mathbb{E}{\left(S(0,{\gamma }_0)\right)}^{1/{\gamma }_0},1)$, we have

$$\sum _{n\ge 1}\mathbb{P}(r_n\ge {\beta }^n)<\infty .$$

Thus, by the Borel–Cantelli Lemma, with a probability of 1, for large enough n, $r_n\le {\beta }^n$. Similarly, for each $\beta \in {\mathbb{R}}_{+}^{\ast }$ and $\gamma >0$, we may write the inequality

$$\begin{array}{ccc}\hfill \mathbb{P}(r_n^{\prime }\le {\beta }^n)& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{-S_n\mathsf{Y}\left(u\right)}{\beta }^{-n}\le 1\}\right)\le {\beta }^{n\gamma }{\left(\mathbb{E}\left(\sum _{i=1}^N{e}^{\gamma {\varphi }_i}\right)\right)}^n.\hfill \end{array}$$

Since $1<\mathbb{E}\left({\sum }_{i=1}^N{e}^{\gamma {\mathsf{Y}}_i}\right)<\infty$, we choose $\beta \in (0,1)$ small enough, so that

$${\beta }^{\gamma }\mathbb{E}\left(\sum _{i=1}^N{e}^{\gamma {\mathsf{Y}}_i}\right)<1,$$

yields that, with a probability of 1 and a large enough n, $r_n^{\prime }\ge {\beta }^n$. Now, using (18) and (17), we get

$$\begin{array}{ccc}\hfill \sum _{|u_j|=e^{-n-j}}{e}^{q{S}_{|u_j|}\mathsf{X}\left(u_j\right)-t{S}_{|u_j|}\mathsf{Y}\left(u_j\right)}& =& \sum _{|u_j|=e^{-n-j}}{e}^{q{S}_{n+j}\mathsf{X}\left(u_j\right)-t{S}_{|n+j|}\mathsf{Y}\left(u_j\right)}\hfill \\ & \ge & e^{-(n+j)ϵln{C}_2}{e}^{-iϵ}(1-e^{-ϵ})=e^{-nϵln{C}_2}{e}^{-jϵ(1+ln{C}_2)}(1-e^{-ϵ})\hfill \\ & \ge & C_3{\left({\displaystyle \frac{1}{C_2^{ϵ}}}\right)}^n(1-e^{-ϵ}),\hfill \end{array}$$

where $C_3$ is a positive constant. Therefore, we conclude that ${\mathcal{P}}_{\chi }^{\ast q,t}\left(A\right)=+\infty$ and ${\Delta }_{\chi }^{\ast q}\left(A\right)\ge t$.

3.

Let ${\displaystyle t>f\left(q\right)≔\inf \left\{t\in \mathbb{R},\underset{n\to \infty }{lim\; sup}\frac{1}{n}\log \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(\left[u\right]\right)-t{S}_n\mathsf{Y}\left(\left[u\right]\right)}\le 0\right\}.}$ Then, there exists $n_0\in \mathbb{N}$ such that

$${\displaystyle \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(\left[u\right]\right)-t{S}_n\mathsf{Y}\left(\left[u\right]\right)}\le 1,n\ge n_0.}$$

It follows that

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{\chi ,n}^{\ast q,t}(\partial \mathsf{T})& =& {\displaystyle \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(\left[u\right]\right)-t{S}_n\mathsf{Y}\left(\left[u\right]\right)}\le 1}\hfill \end{array}$$

and then ${\mathcal{P}}_{\chi }^{\ast q,t}<\infty .$ This implies that ${\Lambda }_{\chi }\left(q\right)\le f\left(q\right)$. On the other hand, assume that $t<f\left(q\right)$, then there exists a sequence ${\left(n_k\right)}_{k\ge 1}$ such that

$${\displaystyle \sum _{u\in {\mathsf{T}}_{n_k}}{e}^{q{S}_n\mathsf{X}\left(\left[u\right]\right)-t{S}_n\mathsf{Y}\left(\left[u\right]\right)}>1.}$$

Therefore,

$$\begin{array}{ccc}\hfill {\mathcal{P}}_{\chi ,n_k}^{\ast q,t}(\partial \mathsf{T})& =& {\displaystyle \sum _{u\in {\mathsf{T}}_{n_k}}{e}^{q{S}_{\left|u\right|}\mathsf{X}\left(u\right)-t{S}_{\left|u\right|}\mathsf{Y}\left(u\right)}>0}\hfill \end{array}$$

and then ${\mathcal{P}}_{\chi }^{\ast q,t}>0.$ This implies that ${\Lambda }_{\chi }\left(q\right)\ge f\left(q\right)$, as required.

□

Remark 3.

In general settings, the assertion 2 of Lemma 2 is not true and needs some some control on studied functions or measures [26,31,32]. In addition, one has ${\Lambda }_{\chi }\left(q\right)={\Lambda }_{\chi }^{\ast }\left(q\right)≔{\Delta }_{\chi }^{\ast q}(\partial \mathsf{T})$.

2.2. Basic Properties of Mandelbrot Measures

We consider the random vector ${\rm Y}≔(N,(X_1,X_1^{\prime }),(X_2,X_2^{\prime }),\dots )$ taking values in ${\mathbb{N}}_{+}\times {(\mathbb{R}\times \mathbb{R})}^{{\mathbb{N}}_{+}}$. Assume that the random variable N satisfies the assumptions above and that

$$\left\{\begin{array}{c}\mathbb{E}\left({\sum }_{i=1}^N\exp \left(X_i^{\prime }\right)\right)=1,\hfill \\ \mathbb{E}\left({\sum }_{i=1}^N{X}_i^{\prime }\exp \left(X_i^{\prime }\right)\right)<0,\hfill \\ \mathbb{E}\left(\left({\sum }_{i=1}^N\exp \left(X_i^{\prime }\right)\right){\log }^{+}\left({\sum }_{i=1}^N\exp \left(X_i^{\prime }\right)\right)\right)<\infty \hfill \end{array}\right.,$$

(19)

$$\mathbb{E}\left(\sum _{i=1}^N\left|X_i\right|\exp \left(X_i^{\prime }\right)\right)<\infty .$$

(20)

and

$$\exists \gamma >1,\mathrm{such}\mathrm{that}\mathbb{E}\left(\right|\sum _{i=1}^N\exp \left(X_i^{\prime }\right){|}^{\gamma })<\infty .$$

(21)

Now, we consider the family $\tilde{\mathrm{Y}}≔{\left\{(N_u,(X_{u1},X_{u1}^{\prime }),(X_{u2},X_{u2}^{\prime }),\dots )\right\}}_u$ of independent copies of $\mathrm{Y}$ indexed by the finite sequences $u=u_1\cdots u_n$, $n\ge 0$, $u_i\in {\mathbb{N}}_{+}$ and defined on a probability space $({\Omega }^{\prime },{\mathcal{A}}^{\prime },{\mathbb{P}}^{\prime })$. Now, condition (19) implies that, with a probability of 1, for all $n\ge 1$ and $u\in {\mathbb{N}}_{+}^n$, the sequence

$$Y_p^{\prime }\left(u\right)=\sum _{v\in {\mathsf{T}}_p\left(u\right)}\exp (S_{n+p}{X}^{\prime }\left(uv\right)-S_n{X}^{\prime }\left(u\right))(p\ge 1)$$

converges to a positive limit $Y^{\prime }\left(u\right)$ whenever the condition is satisfied, whereas the limit exists but equals zero if the condition fails. This result was established in [33] for the case of constant N and extended in [34] to the general setting. Using the family $\Pi$, one can then construct the Mandelbrot measure on the $\sigma$-algebra $\mathcal{C}$ generated by the cylinders of ${\mathbb{N}}_{+}^{{\mathbb{N}}_{+}}$, defined as

$${\mu }^{\prime }\left(\left[u\right]\right)=\left\{\begin{array}{cc}\exp \left(S_n{X}^{\prime }\left(u\right)\right)Y^{\prime }\left(u\right)\hfill & \mathrm{if}u\in {\mathsf{T}}_n\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

and supported on $\partial \mathsf{T}$. The branching measure corresponds to the choice $X_i^{\prime }=-\log \mathbb{E}\left(N\right)$ for $1\le i\le N$. The next theorem is crucial in our study and will be exploited to construct an adequate Mandelbrot measure $\mu$ in Section 4.2 supported by the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$. Moreover, we will prove that this measure is exact dimensional with dimension D in the sense that $D=\inf \left\{\dim F:F\mathrm{Borel},\mu \left(F\right)>0\right\}=\inf \left\{\mathrm{Dim}F:F\mathrm{Borel},\mu \left(F\right)=\parallel \mu \parallel \right\}$.

Theorem 1.

• With a probability of 1 for ${\mu }^{\prime }$, almost every (a.e.) $\mathsf{t}\in \partial \mathsf{T}$,

  $$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\log Y^{\prime }\left({\mathsf{t}}_{|n}\right)}{-n}}\le 0\mathit{and}\mathit{then}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\log {\mu }^{\prime }\left(\left[{\mathsf{t}}_{|n}\right]\right)}{-n}}\le -\mathbb{E}\left(\sum _{i=1}^N{X}_i^{\prime }\exp \left(X_i^{\prime }\right)\right).$$
• Assume (20). Then, with a probability of 1, for ${\mu }^{\prime }$, almost every (a.e.) $\mathsf{t}\in \partial \mathsf{T}$,

  $$\underset{n\to \infty }{\lim }{\displaystyle \frac{S_{n}X\left(\mathsf{t}\right)}{n}}=\mathbb{E}\left(\sum _{i=1}^N{X}_i\exp \left(X_i^{\prime }\right)\right).$$
• Assume (21). Then, with a probability of 1, for ${\mu }^{\prime }$, almost every $\mathsf{t}\in \partial \mathsf{T}$,

  $$\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{\log {\mu }^{\prime }\left(\left[{\mathsf{t}}_{|n}\right]\right)}{-n}}\ge -\mathbb{E}\left(\sum _{i=1}^N{X}_i^{\prime }\exp \left(X_i^{\prime }\right)\right).$$

The first assertion is due to the fact that $E\left(Y^{\prime }\right)<\infty$. The second assertion is an application of the the strong law of large numbers to $S_{n}X\left(t\right)$ with respect to Peyrière’s measure ${\mathcal{Q}}^{\prime }$ defined on ${\mathcal{A}}^{\prime }\otimes \mathcal{C}$ as ${\mathcal{Q}}^{\prime }\left(E\right)={\mathbb{E}}_{{\mathbb{P}}^{\prime }}(\int {\mathbf{1}}_E({\omega }^{\prime },t){\mu }^{\prime }\left(\mathrm{d}\mathsf{t}\right))$ [33,35,36]. The last assertion holds under the property $\mathbb{E}\left(Y^{\prime }{\log }^{+}{Y}^{\prime }\right)<\infty$ and in particular when $\mathbb{E}\left({Y^{\prime }}^h\right)<\infty$ for some $h>1$ (see again [33,35,36]). This is true under (21). The reader can see the multifractal analysis of Mandelbrot measures in [28,37,38,39,40] (under always weaker assumptions). We end this section by the mass distribution principle, which will be used to estimate the lower bound of the Hausdorff dimension of the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$ in Section 4.2.

Theorem 2.

([9]). Let ν be a positive and finite Borel probability measure on a compact metric space $(X,d)$. Assume that $M\subseteq X$ is a Borel set such that $\nu \left(M\right)>0$ and

$${\displaystyle M\subseteq \left\{s\in X,\underset{r\to 0^{+}}{lim\; inf}\frac{\log \nu \left(B\right(s,r\left)\right)}{\log r}\ge \delta \right\}.}$$

Then, the Hausdorff dimension of M is bounded from below by δ.

3. Study of Sets ${\mathrm{E}}_{\mathbf{\chi }}\left(\mathit{\alpha }\right)$ and ${\mathrm{E}}_{\mathbf{\chi }}(\mathit{\alpha },\mathit{\beta })$ Under Metric ${\mathit{d}}_{\mathbf{1}}$

In this section, we will study the multifractal analysis of branching random walks when $\partial \mathsf{T}$ is endowed with the metric $d_1$. We define, for all $(q,t)\in {\mathbb{R}}^2$,

$$S(q,t)=\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i-t{\mathsf{Y}}_i\right)$$

and let $\tau \left(q\right)$ be a unique t such that $\mathbb{E}\left(S(q,\tau (q)\right)=1$. We assume

$$\forall (q,\alpha )\in {\mathbb{R}}^2,\mathrm{such}\mathrm{that}{\tau }^{\ast }\left({\tau }^{\prime }\left(q\right)\right)>0,\exists \gamma >1,\mathbb{E}\left(S{(q,\tau \left(q\right))}^{\gamma }\right)<\infty .$$

(22)

and

$$\underset{q\in \mathbb{R}}{\sup }\mathbb{E}\left(\sum _{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i\right)<\infty .$$

(23)

Lemma 3.

For all $q\in \mathbb{R}$, we have ${\Lambda }_{\chi }\left(q\right)\le \tau \left(q\right)$, when ${\Lambda }_{\chi }$ given in Lemma 2.

Proof. 

Let $t>\tau \left(q\right)$ and recall

$${\displaystyle {\Lambda }_{\chi }\left(q\right)=\inf \left\{s\in \mathbb{R},\underset{n\to \infty }{lim\; sup}\frac{1}{n}\log \sum _{u\in {\mathsf{T}}_n}{e}^{q{S}_n\mathsf{X}\left(u\right)-s{S}_n\mathsf{Y}\left(u\right)}\le 0\right\}.}$$

Then,

$$\begin{array}{ccc}\hfill {\displaystyle \mathbb{E}\left(\sum _{n\ge 1}\sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right))\right)}& =& {\displaystyle \sum _{n\ge 1}\mathbb{E}{\left(\sum _{i=1}^N\exp (q{\mathsf{X}}_i-t{\mathsf{Y}}_i)\right)}^n<\infty .}\hfill \end{array}$$

It follows that ${\displaystyle \sum _{u\in {\mathsf{T}}_n}\exp \left(q{S}_n\mathsf{X}\left(u\right)-t{S}_n\mathsf{Y}\left(u\right)\right)}$ is bounded a.s., so $t\ge {\Lambda }_{\chi }\left(q\right)$ a.s. Since $t>\tau \left(q\right)$ is arbitrary, we have the conclusion. □

Let $I=\{\alpha \in \mathbb{R},{\mathrm{E}}_{\chi }\left(\alpha \right)\ne \varnothing \}$. With the same line as in [25], Proposition 3.2, and using (23), we have $int\left(I_{\chi }\right)=\left\{{\tau }^{\prime }\left(q\right),{\tau }^{\ast }\left({\tau }^{\prime }\left(q\right)\right)>0\right\}$. We will prove, in this section, the following result.

Theorem 3.

Assume (4), (22) and (23) Then, for every $\alpha \in int\left(I_{\chi }\right)$, we have a.s.

$$\dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)=\mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)=\beta {\tau }^{\ast }\left(\alpha \right),$$

where $\beta =\mathbb{E}\left({\sum }_{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i)\right).$

In general settings, one can estimate the upper bounds of the (standard) Hausdorff and packing dimensions $\dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)$ and $\mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)$. Indeed, using Lemma 1, we have

• If ${\mathsf{B}}_{\chi }\left(q\right)-q\alpha \ge 0$, then $\mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta \left({\mathsf{B}}_{\chi }\left(q\right)-q\alpha \right).$
• If ${\mathsf{b}}_{\chi }\left(q\right)-q\alpha \ge 0$, then $\dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta \left({\mathsf{b}}_{\chi }\left(q\right)-q\alpha \right).$

Hence, for all $\alpha \in \mathbb{R}$ such that ${\Lambda }_{\chi }\left(q\right)-q\alpha \ge 0$, we have

$${\displaystyle \dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta \left({\Lambda }_{\chi }\left(q\right)-q\alpha \right),}$$

(24)

where a negative dimension means that ${\mathrm{E}}_{\chi }(\alpha ,\beta )$ is empty. In particular, if $\alpha ={\Lambda }_{\chi }^{\prime }\left(q\right)$, then ${\displaystyle \dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)\le \beta {\Lambda }_{\chi }^{\ast }\left(\alpha \right).}$ In what follows, we give a sufficient condition to obtain the lower bound of Hausdorff dimension.

Proposition 1.

Let $\alpha ={\Lambda }_{\chi }^{\prime }\left(q\right)$ and assume that ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left({\mathrm{E}}_{\chi }\left({\Lambda }_{\chi }^{\prime }\left(q\right)\right)\right)>0$. Then,

$$\dim \left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)=\mathrm{Dim}\left({\mathrm{E}}_{\chi }(\alpha ,\beta )\right)=\beta b_{\chi }^{\ast }\left(\alpha \right)=\beta B_{\chi }^{\ast }\left(\alpha \right)=\beta {\Lambda }_{\chi }^{\ast }\left(\alpha \right).$$

Proof. 

For $\epsilon >0$, if $m\ge n$, we define the set

$$A_m\left(\epsilon \right)=\left\{\mathsf{t}\in {\mathrm{E}}_{\chi }\left(\alpha \right)\left|\right|S_n\mathsf{X}\left(\mathsf{t}\right)-{\Lambda }_{\chi }^{\prime }\left(q\right)S_n\mathsf{Y}\left(\mathsf{t}\right)|\le \epsilon S_n\mathsf{Y}\left(\mathsf{t}\right)\mathrm{for}n\ge m\right\}.$$

Since ${\displaystyle {\mathrm{E}}_{\chi }\left(\alpha \right)=\bigcup _{m\ge 1}{A}_m\left(\epsilon \right)}$, then there exists $m\ge 1$ such that ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(A_m\left(\epsilon \right)\right)>0$. Let $\left\{\left[u_i\right]\right\}$ be an $e^{-l}$ covering of $A_m\left(\epsilon \right)$. Then, for $q\ge 0$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \sum e^{-\left({\Lambda }_{\chi }\left(q\right)-q{\Lambda }_{\chi }^{\prime }\left(q\right)-q\epsilon \right)S_n\mathsf{Y}\left(u_i\right)}}& \ge & {\displaystyle \sum e^{q{S}_n\mathsf{X}\left(u_i\right)-{\Lambda }_{\chi }\left(q\right)S_n\mathsf{Y}\left(u_i\right)}}\hfill \\ & \ge & {\mathcal{H}}_{\chi ,e^{-l}}^{q,{\Lambda }_{\chi }\left(q\right)}\left(A_m\left(\epsilon \right)\right).\hfill \end{array}$$

Hence, ${\mathcal{H}}_{\mathsf{Y}}^{{\Lambda }_{\chi }\left(q\right)-q{\Lambda }_{\chi }^{\prime }\left(q\right)-q\epsilon }\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)\ge {\mathcal{H}}_{\mathsf{Y}}^{{\Lambda }_{\chi }\left(q\right)-q{\Lambda }_{\chi }^{\prime }\left(q\right)-q\epsilon }\left(A_m\left(\epsilon \right)\right)\ge {\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left(A_m\left(\epsilon \right)\right)>0$. This implies that

$${\dim }_{\mathsf{Y}}\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)\ge {\Lambda }_{\chi }\left(q\right)-q{\Lambda }_{\chi }^{\prime }\left(q\right)-q\epsilon .$$

Similarly, if $q<0$, one has ${\dim }_{\mathsf{Y}}\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)\ge {\Lambda }_{\chi }\left(q\right)-q{\Lambda }_{\chi }^{\prime }\left(q\right)+q\epsilon .$ Therefore, the result follows by letting $\epsilon \to 0$. □

Remark 4.

• In general ${\Lambda }_{\chi }$ is not differentiable, but here ${\Lambda }_{\chi }=\tau$, and then it is differentiable.
• Assume that ${\mathcal{H}}_{\chi }^{q,\tau \left(q\right)}\left({\mathrm{E}}_{\chi }\left({\tau }_{\chi }^{\prime }\left(q\right)\right)\right)>0$, then ${\mathcal{H}}_{\chi }^{q,{\Lambda }_{\chi }\left(q\right)}\left({\mathrm{E}}_{\chi }\left({\tau }_{\chi }^{\prime }\left(q\right)\right)\right)>0$ by Lemma (3). In addition, if ${\tau }_{\chi }^{\prime }\left(q\right)\ne {\Lambda }_{\chi }^{\prime }\left(q\right)$, then ${\mathrm{E}}_{\chi }\left({\tau }_{\chi }^{\prime }\left(q\right)\right)\bigcap {\mathrm{E}}_{\chi }\left({\Lambda }_{\chi }^{\prime }\left(q\right)\right)=\varnothing$, which contradicts Lemma 1. Therefore, we necessarily have ${\mathcal{H}}_{\chi }^{q,\Lambda \left(q\right)}\left({\mathrm{E}}_{\chi }\left({\Lambda }_{\chi }^{\prime }\left(q\right)\right)\right)>0$. Therefore, we have ${\Lambda }_{\chi }=\tau$.

First, observe that

$${\tau }^{\prime }\left(q\right)={\displaystyle \frac{\mathbb{E}\left({\sum }_{i=1}^N{\mathsf{X}}_i\exp \left(q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i\right)\right)}{\mathbb{E}\left({\sum }_{i=1}^N{\mathsf{Y}}_i\exp \left(q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i\right)\right)}}.$$

(25)

For each $(q,\alpha )\in {\mathbb{R}}^2$, a Mandelbrot measure ${\mu }_q$ on $\partial \mathsf{T}$ is associated with the vectors $(N_u,\exp (q{\mathsf{X}}_{u1}-\tau \left(q\right){\mathsf{Y}}_{u1}),\exp (q{\mathsf{X}}_{u2}-\tau \left(q\right){\mathsf{Y}}_{u2}),\dots )$, $u\in {\bigcup }_{n\ge 0}{\mathbb{N}}_{+}^n$. Moreover, the measure ${\mu }_{q,\alpha }$ is non-degenerate if and only if

$$-\mathbb{E}\left(\sum _{i=1}^N\left(q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i\right)\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i)\right)>0.$$

Since $\mathbb{E}\left({\sum }_{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i)\right)>0$, we have that ${\mu }_q$ is non-degenerate if and only if

$${\tau }^{\ast }\left({\tau }^{\prime }\left(q\right)\right)=\tau \left(q\right)-q{\tau }^{\prime }\left(q\right)>0.$$

(26)

In addition (see Section 2.2), with a probability of 1, we have ${\mu }_q$ a.e.

$$\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{n}}={\alpha }_{\mathsf{X}}\left(q\right)\mathrm{and}\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}={\alpha }_{\mathsf{Y}}\left(q\right),$$

(27)

where

$${\alpha }_{\mathsf{X}}\left(q\right)=\mathbb{E}\left(\sum _{i=1}^N{\mathsf{X}}_i\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i)\right)$$

$${\alpha }_{\mathsf{Y}}\left(q\right)=\mathbb{E}\left(\sum _{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i-\tau \left(q\right){\mathsf{Y}}_i)\right).$$

Let $\alpha \in int\left(I_{\chi }\right)$ and construct the Mandelbrot measure ${\mu }_{q_{\alpha }}$ on $\partial \mathsf{T}$, where $q_{\alpha }$ is defined in Lemma 5 such that ${\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)\right)>0$, and ${\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)=0$; that is, using (30) and (33),

$${\alpha }_{\mathsf{X}}\left(q_{\alpha }\right)=\alpha {\alpha }_{\mathsf{Y}}\left(q_{\alpha }\right).$$

The fact that ${\tau }^{\ast }\left({\tau }^{\prime }\left(q_{\alpha }\right)\right)>0$ implies that this measure is a.s. positive by (32). Moreover, with a probability of 1, we have

$$\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}={\displaystyle \frac{{\alpha }_{\mathsf{X}}\left(q_{\alpha }\right)}{{\alpha }_{\mathsf{Y}}\left(q_{\alpha }\right)}}=\alpha {\mu }_{q_{\alpha }}\text{-}\mathrm{a}.\mathrm{e}..$$

(28)

It follows that ${\mu }_{q_{\alpha }}\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)=1$. Moreover, there exists $C>0$ such that, for all $t\in {\mathrm{E}}_{\chi }\left(\alpha \right)$, one has

$$\begin{array}{c}\hfill {\mu }_{q_{\alpha }}\left(\left[{\mathsf{t}}_{|n}\right]\right)\le C{e}^{q_{\alpha }{S}_n\mathsf{X}\left(\left[t_{|n}\right]\right)-\tau \left(q_{\alpha }\right)S_n\mathsf{Y}\left(\left[t_{|n}\right]\right)},\end{array}$$

for large enough n. This implies (Frostman Lemma [31]) that ${\mathcal{H}}_{\chi }^{q_{\alpha },\tau \left(q_{\alpha }\right)}\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)>0$. Therefore for $\alpha ={\Lambda }_{\chi }^{\prime }\left(q\right)$, one has ${\mathcal{H}}_{\chi }^{q_{\alpha },{\Lambda }_{\chi }\left(q_{\alpha }\right)}\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)>0$ from Remark 4. In addition, using Lemma 1, we obtain $b_{\chi }\left(q_{\alpha }\right)=B_{\chi }\left(q_{\alpha }\right)={\Lambda }_{\chi }\left(q_{\alpha }\right)=\tau \left(q_{\alpha }\right).$ The result therefore follows from Proposition 1.

Remark 5.

In fact, since we explicitly know the Mandelbrot measure, one can compute the dimension of measure  ${\mu }_q$ and obtain a conclusion using the mass distribution principle. More precisely, by (22) and Theorem 1, we have a.s. for ${\mu }_q$ almost every t,

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{\lim }{\displaystyle \frac{\log {\mu }_{q_{\alpha }}\left(\left[{\mathsf{t}}_{|n}\right]\right)}{\log \left(\right|{\mathsf{t}}_{|n}\left|\right)}}& =& -\mathbb{E}\left(\sum _{i=1}^N(q_{\alpha }{\mathsf{X}}_i-\tau \left(q_{\alpha }\right){\mathsf{Y}}_i)\exp (q_{\alpha }{\mathsf{X}}_i-\tau \left(q_{\alpha }\right){\mathsf{Y}}_i)\right)\hfill \\ & =& \tau \left(q_{\alpha }\right){\alpha }_{\mathsf{Y}}\left(q_{\alpha }\right)-q_{\alpha }{\alpha }_{\mathsf{X}}\left(q_{\alpha }\right)\hfill \\ & {=}^{\left(34\right)}& \left(\tau \left(q_{\alpha }\right)-q_{\alpha }{\tau }^{\prime }\left(q_{\alpha }\right)\right){\alpha }_{\mathsf{Y}}\left(q_{\alpha }\right)=\beta {\tau }^{\ast }\left({\tau }^{\prime }\left(q_{\alpha }\right)\right).\hfill \end{array}$$

where we used the fact that $\beta ={\alpha }_{\mathsf{Y}}\left(q_{\alpha }\right)$. Therefore, the Mandelbrot measure ${\mu }_{q_{\alpha }}$ is exact dimensional with dimension $\beta {\tau }^{\ast }\left({\tau }^{\prime }\left(q_{\alpha }\right)\right).$ Finally, we deduce the result by the mass distribution principle (Theorem 2).

4. Study of Sets ${\mathrm{E}}_{\mathbf{\chi }}\left(\mathit{\alpha }\right)$ and ${\mathrm{E}}_{\mathbf{\chi }}(\mathit{\alpha },\mathit{\beta })$ Under Metric ${\mathit{d}}_{\Phi }$

In this section, we will study these sets when $\partial \mathsf{T}$ is endowed with the distance $d_{\Phi }$. This generalizes the results proven in [22,23,24]. Recall (18), Then, with a probability of 1, $S_n\mathsf{Y}\left(u\right)$ tends to ∞ uniformly in $u\in {\mathsf{T}}_n$, as $n\to \infty$. It follows, under (4),

$${\mathrm{E}}_{\chi }\left(\alpha \right)=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\left(\mathsf{X}\left(\mathsf{t}\right)-\alpha \mathsf{Y}\left(\mathsf{t}\right)\right)}{n}}=0\right\}.$$

We will assume that

$$\sup \mathbb{E}\left(\sum _{i=1}^N{\Phi }_i\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-\Pi (q,\alpha )\right)\right)<\infty ,$$

(29)

where the supremum is over all $q\in \mathbb{R}$ and $\alpha \in int\left(I_{\chi }\right).$ Recall the definition of the function ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$ in Section 1. Then, for ${\Phi }_i=1$ for all $1\le i\le N$, ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)=\Pi (q,\alpha )$. Moreover, if ${\Phi }_i={\mathsf{Y}}_i=1$ for all $1\le i\le N$, then

$${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)=\tilde{P}\left(q\right)-q\alpha \mathrm{and}{\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right)={\tilde{P}}^{\ast }\left(\alpha \right).$$

Lemma 4.

Assume (4). For every $n\ge 1$ and $\beta >0$, let us denote

$$\left\{\begin{array}{cc}{D}_n^{\beta }=\max \left\{\mathrm{diam}\left(\left[u\right]\right)e^{\beta S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\right\}\hfill & \\ d_n^{\beta }=\min \left\{\mathrm{diam}\left(\left[u\right]\right)e^{\beta S_n\mathsf{Y}\left(u\right)}:u\in {\mathsf{T}}_n\right\}.\hfill \end{array}\right.$$

• There exits ${\beta }_1>0$ such that, with a probability of 1, $d_n^{{\beta }_1}\ge 1$ for large enough n.
• There exits ${\beta }_0>0$ such that, with a probability of 1, $D_n^{{\beta }_0}\le 1$, for large enough n.

Proof. 

1.

For each $\beta >0$ and $\gamma >0$, we have

$$\begin{array}{ccc}\hfill \mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\ge \beta \})& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{S_n\Phi \left(u\right)}\ge e^{\beta S_n\mathsf{Y}\left(u\right)}\right)=\mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{S_n\Phi \left(u\right)-\beta S_n\mathsf{Y}\left(u\right)}\ge 1\right)\hfill \\ & =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{\gamma S_n\Phi \left(u\right)-\gamma \beta S_n\mathsf{Y}\left(u\right)}\ge 1\right)\hfill \end{array}$$

Notice, under (4), that ${\Phi }_i\le M_{\Phi }$ a.s. for each $1\le i\le N$ for some positive real $M_{\Phi }$. Therefore,

$$\mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\ge \beta \})\le \mathbb{E}\left(\sum _{u\in {\mathsf{T}}_n}{e}^{\gamma (S_n\Phi \left(u\right)-\beta S_n\mathsf{Y}\left(u\right))}\right)\le e^{n\gamma M_{\Phi }}{\left(\mathbb{E}\left(\sum _{i=1}^N{e}^{-\beta \gamma {\mathsf{Y}}_i}\right)\right)}^n.$$

We can choose $\beta$ large enough such that $e^{\gamma M_{\Phi }}\mathbb{E}\left({\sum }_{i=1}^N{e}^{-\beta \gamma {\mathsf{Y}}_i}\right)<1$. Now, observe that

$$\begin{array}{ccc}\hfill \mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\ge \beta \})& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,S_n\Phi \left(u\right))\ge \beta S_n\mathsf{Y}\left(u\right)\}\right)\hfill \\ & =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{-S_n\Phi \left(u\right))}\le e^{-\beta S_n\mathsf{Y}\left(u\right)}\}\right)\hfill \\ & =& \mathbb{P}\left(d_n^{\beta }\le 1\}\right).\hfill \end{array}$$

It follows that ${\sum }_{n\ge 1}\mathbb{P}\left(d_n^{\beta }\le 1\right)<\infty$; hence, by the Borel–Cantelli Lemma, with a probability of 1 for large enough n, $d_n^{\beta }\ge 1$.

2.

For each $\beta >0$ and $\gamma >0$, we have

$$\begin{array}{ccc}\hfill \mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\le \beta \})& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{S_n\Phi \left(u\right)}\le e^{\beta S_n\mathsf{Y}\left(u\right)}\right)\hfill \\ & =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{-\gamma S_n\Phi \left(u\right)+\gamma \beta S_n\mathsf{Y}\left(u\right)}\ge 1\right)\hfill \\ & \le & \mathbb{E}\left(\sum _{u\in {\mathsf{T}}_n}{e}^{-\gamma (S_n\Phi \left(u\right)-\beta S_n\mathsf{Y}\left(u\right))}\right)\le {\left(\mathbb{E}\left(\sum _{i=1}^N{e}^{-\gamma ({\Phi }_i-\beta {\mathsf{Y}}_i}\right)\right)}^n.\hfill \end{array}$$

Notice, under (4), that ${\mathsf{Y}}_i\le M_{\mathsf{Y}}$ a.s. for each $1\le i\le N$ and for some positive real $M_{\mathsf{Y}}$. Therefore,

$$\mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\le \beta \})\le e^{n\gamma \beta M_{\mathsf{Y}}}(\mathbb{E}{\left(\sum _{i=1}^N{e}^{-\gamma {\Phi }_i}\right)}^n.$$

We can choose $\gamma$ such that $(\mathbb{E}\left({\sum }_{i=1}^N{e}^{-\gamma {\Phi }_i}\right)<1$. Now, choose ${\beta }_0$ small enough so that $e^{\gamma {\beta }_0{M}_{\mathsf{Y}}}(\mathbb{E}\left({\sum }_{i=1}^N{e}^{-\gamma {\Phi }_i}\right)<1$. It follows that

$$\begin{array}{ccc}\hfill \mathbb{P}(\{\exists u\in {\mathsf{T}}_n,{\displaystyle \frac{S_n\Phi \left(u\right))}{S_n\mathsf{Y}\left(u\right)}}\le {\beta }_0\})& =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,S_n\Phi \left(u\right))\le {\beta }_0{S}_n\mathsf{Y}\left(u\right)\}\right)\hfill \\ & =& \mathbb{P}\left(\{\exists u\in {\mathsf{T}}_n,e^{-S_n\Phi \left(u\right))}\ge e^{-{\beta }_0{S}_n\mathsf{Y}\left(u\right)}\}\right)\hfill \\ & =& \mathbb{P}\left(D_n^{{\beta }_0}\ge 1\}\right).\hfill \end{array}$$

It follows that ${\sum }_{n\ge 1}\mathbb{P}\left(D_n^{{\beta }_0}\ge 1\}\right)<\infty$; hence, by the Borel–Cantelli Lemma, with a probability of 1, for large enough n, $D_n^{{\beta }_0}\ge 1$.

□

Corollary 1.

Assume (4), then there exists ${\beta }_0,{\beta }_1>0$ such that

$${\beta }_0\le {\displaystyle \frac{S_n\Phi \left(u\right)}{S_n\mathsf{Y}\left(u\right)}}\le {\beta }_1.$$

Our main result in this section is the following:

Theorem 4.

Assume (4), (7), (8), and (29). If $\partial \mathsf{T}$ is endowed with the random ultrametric distance $d_{\Phi }$, then, with probability 1, for every α such that ${\inf }_{q\in \mathbb{R}}\Pi (q,\alpha )>0$, we have

$${\dim }_{\Phi }{\mathrm{E}}_{\chi }\left(\alpha \right)={\mathrm{Dim}}_{\Phi }{\mathrm{E}}_{\chi }\left(\alpha \right)=\underset{q\in \mathbb{R}}{\inf }{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)={\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right).$$

Corollary 2.

Assum, (7), (8), and (4) for $\mathsf{Z}=\mathsf{Y}$ are satisfied. If $\partial \mathsf{T}$ is endowed with the standard distance $d_1$, then, with probability 1, for every α such that ${\inf }_{q\in \mathbb{R}}\Pi (q,\alpha )>0$, we have

$$\dim {\mathrm{E}}_{\chi }\left(\alpha \right)=\mathrm{Dim}{\mathrm{E}}_{\chi }\left(\alpha \right)=\underset{q\in \mathbb{R}}{\inf }\Pi (q,\alpha ).$$

Proof. 

This is a direct consequence of Theorem 4 for ${\Phi }_i=1$ for all $1\le i\le N$. Indeed, in this case, (4) is satisfied since $\mathbb{E}\left(N\right)<\infty$. Moreover, by definition of $\Pi$, one has

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\sum _{i=1}^N{\Phi }_i\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-\Pi (q,\alpha )\right)\right)& =& \exp \left(-\Pi (q,\alpha )\right)\mathbb{E}(\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i\right)\hfill \\ & =& 1<\infty ,\hfill \end{array}$$

so that $\left(29\right)$ is satisfied. □

4.1. Upper Bound of Sets ${\mathrm{E}}_{\chi }\left(\alpha \right)$ Under Metric $d_{\Phi }$

For each $(q,\alpha ,t)\in {\mathbb{R}}^3$, let us define

$${\displaystyle P_{\alpha }\left(q\right)=\inf \left\{t\in \mathbb{R}:\underset{n\to \infty }{lim\; sup}\frac{1}{n}\log \left(\sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n(\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(u\right)-t{S}_n\Phi \left(u\right))\right)\le 0\right\}.}$$

Since the mappings ${\displaystyle (q,t)\mapsto \sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(u\right)-t{S}_n\Phi \left(u\right))}$ and ${\displaystyle (\alpha ,t)\mapsto \sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(u\right)-t{S}_n\Phi \left(u\right))}$ for a given $\alpha \in \mathbb{R}$ and $q\in \mathbb{R}$, respectively, we obtain that the mappings $q\mapsto P_{\alpha }\left(q\right)$ and $\alpha \mapsto P_{\alpha }\left(q\right)$ are convex.

Proposition 2.

With a probability of 1,

• $P_{\alpha }\left(q\right)\le {\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$ for all $(q,\alpha )\in {\mathbb{R}}^2$.
• ${\dim }_{\Phi }\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)\le P_{\alpha }^{\ast }\left(0\right)$, for all $\alpha \in \mathbb{R}$.
• ${\mathrm{Dim}}_{\Phi }\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)\le {\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right)$, for all $\alpha \in \mathbb{R}$.

Proof. 

1.

We only need to prove the inequality for each $(q,\alpha )\in {\mathbb{R}}^2$ a.s. Fix $(q,\alpha )\in {\mathbb{R}}^2$. For $t>{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \mathbb{E}(\sum _{n\ge 1}\sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n(\mathsf{X}-\alpha \mathsf{Y})\left(u\right)-t{S}_n\Phi \left(u\right))}& =& {\displaystyle \sum _{n\ge 1}\mathbb{E}{(\sum _{i=1}^N\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-t{\Phi }_i))}^n<\infty .}\hfill \end{array}$$

It follows that ${\displaystyle \sum _{u\in {\mathsf{T}}_n}\exp \left(q{S}_n(\mathsf{X}-\alpha \mathsf{Y})\left(u\right)-t{S}_n\Phi \left(u\right)\right)}$ is bounded a.s., so $t\ge P_{\alpha }\left(q\right)$ a.s. Since $t>{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$ is arbitrary, we have the conclusion.

2.

For every $n\ge 1$, let us denote $r_n=\max \{\exp (-S_n\Phi \left(u\right)):u\in {\mathsf{T}}_n\}$. Recall that

$$\begin{array}{ccc}\hfill {\mathrm{E}}_{\chi }\left(\alpha \right)& =& \bigcap _{ϵ>0}\bigcap _{N\ge 1}\bigcup _{n\ge N}\{\mathsf{t}\in \partial \mathsf{T}:|S_n\mathsf{X}\left(\mathsf{t}\right)-\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)|\le ϵS_n\mathsf{Y}\left(\mathsf{t}\right)\}.\hfill \end{array}$$

Fix $q\ge 0$ and $ϵ>0$. For $N\ge 1$, the set $E(q,N,ϵ,\alpha )={\bigcup }_{n\ge N}\{\mathsf{t}\in \partial \mathsf{T}:q{S}_n\mathsf{X}\left(\mathsf{t}\right)-\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)v\le qϵS_n\mathsf{Y}\left(\mathsf{t}\right)\}$ is covered by the union of those $\left[u\right]$ such that $u\in {\mathsf{T}}_n$ and $q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)+qϵS_n\mathsf{Y}\left(u\right)\ge 0$. Consequently, for $s\ge 0$,

$$\begin{array}{cc}\hfill {\mathcal{H}}_{\Phi ,-\log r_N}^s\left(E(q,N,ϵ,\alpha )\right)& \le \sum _{n\ge N}\sum _{u\in {\mathsf{T}}_n}\exp (-s{S}_n\Phi \left(\mathsf{t}\right))\exp (q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)+qϵS_n\mathsf{Y}\left(\mathsf{t}\right))\hfill \end{array}$$

Hence, if $\eta >0$ and $s>P_{\alpha }\left(q\right)+\eta +qϵ/{\beta }_0$, by definition of $P_{\alpha }\left(q\right)$, for large enough n we have

$$\begin{array}{cc}\hfill {\mathcal{H}}_{\Phi ,-\log r_N}^s\left(E(q,N,ϵ,\alpha )\right)& \le \sum _{n\ge N}\sum _{u\in {\mathsf{T}}_n}\exp (-s{S}_n\Phi \left(\mathsf{t}\right))\exp (q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)+qϵ/{\beta }_0{S}_n\Phi \left(\mathsf{t}\right))\hfill \\ \hfill & \le \sum _{n\ge N}\sum _{u\in {\mathsf{T}}_n}\exp (q{S}_n\mathsf{X}\left(u\right)-q\alpha S_n\mathsf{Y}\left(\mathsf{t}\right)-(P_{\alpha }\left(q\right)-\eta )S_n\Phi \left(\mathsf{t}\right)),\hfill \end{array}$$

which implies that

$${\mathcal{H}}_{\Phi ,-\log r_N}^s\left(E(q,N,ϵ,\alpha )\right)\le \sum _{n\ge N}{e}^{-n\eta /2}.$$

Since $r_N$ tends to 0 a.s. as N tends to ∞, we thus have ${\mathcal{H}}^s\left(E(q,N,ϵ,\alpha )\right)=0$ (since we assume (4)); hence, $\dim E(q,N,ϵ,\alpha )\le s$. Since this holds for all $\eta >0$, we obtain $\dim E(q,N,ϵ,\alpha )\le P_{\alpha }\left(q\right)+qϵ/{\beta }_0$. It follows that

$${\dim }_{\Phi }{\mathrm{E}}_{\chi }\left(\alpha \right)\le \underset{q\ge 0}{\inf }\underset{ϵ>0}{\inf }{P}_{\alpha }\left(q\right)+qϵ/{\beta }_0=\underset{q\ge 0}{\inf }{P}_{\alpha }\left(q\right).$$

Similarly, if $q<0,$ we have ${\dim }_{\Phi }{\mathrm{E}}_{\chi }\left(\alpha \right)\le {\inf }_{q<0}{\inf }_{ϵ>0}{P}_{\alpha }\left(q\right)-qϵ/{\beta }_0={\inf }_{q<0}{P}_{\alpha }\left(q\right).$ In particular, if ${\inf }_{q\in \mathbb{R}}{P}_{\alpha }\left(q\right)<0$, we necessarily have ${\dim }_{\Phi }{\mathrm{E}}_{\chi }\left(\alpha \right)=\varnothing$.

3.

Let $\alpha \in \mathbb{R}$ and $ϵ>0$. We notice that $\mathrm{E}\left(\alpha \right)\subset {\bigcup }_{N\ge 1}{F}_N$, where

$$F_N=\bigcap _{n\ge N}\left\{\mathsf{t}\in \mathsf{T}:|S_n(\mathsf{X}\left(t\right)-\alpha \mathsf{Y}\left(t\right))|\le ϵS_n\mathsf{Y}\left(\mathsf{t}\right)\right\}.$$

Fix $N\ge 1$. Let $u\in \mathsf{T}$ with $\left|u\right|\ge N$. For each $N^{\prime }\ge \left|u\right|$, any $e^{-N^{\prime }}$ packing of $\left[u\right]\cap F_N$ is included in

$$\bigcup _{n\ge N^{\prime }}\bigcup _{v\in {\mathsf{T}}_n,\left|S_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))\right|\le ϵS_n\mathsf{Y}\left(v\right)}\left[v\right].$$

Consequently, for $q>0$, setting $s={\tilde{\mathcal{P}}}_{\alpha }\left(q\right)+ϵq/{\beta }_0+\eta$, we have

$$\begin{array}{ccc}& & \underset{{\left\{\left[v_j\right]\right\}}_{j\in \mathcal{I}},e^{-N^{\prime }}-\mathrm{packing}\mathrm{of}\left[u\right]\cap F_N}{\sup }\sum _{j\in \mathcal{I}}\left|\right[v_j]{|}^s\le \sum _{n\ge N^{\prime }}\sum _{v\in {\mathsf{T}}_n,\left|S_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))\right|\le ϵS_n\mathsf{Y}\left(v\right)}{\left|\left[v\right]\right|}^s\hfill \\ & \le & \sum _{n\ge N^{\prime }}\sum _{v\in {\mathsf{T}}_n}\exp \left(q{S}_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))+ϵq{S}_n\mathsf{Y}\left(v\right)-(s-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right))S_n\Phi \left(v\right)-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)S_n\Phi \left(v\right)\right).\hfill \end{array}$$

Now, using Lemma 4, we obtain for large enough n

$$\begin{array}{ccc}& & \underset{{\left\{\left[v_j\right]\right\}}_{j\in \mathcal{I}},e^{-N^{\prime }}\mathrm{packing}\mathrm{of}\left[u\right]\cap F_N}{\sup }\sum _{j\in \mathcal{I}}\left|\right[v_j]{|}^s\le \sum _{n\ge N^{\prime }}\sum _{v\in {\mathsf{T}}_n,\left|S_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))\right|\le ϵS_n\mathsf{Y}\left(v\right)}{\left|\left[v\right]\right|}^s\hfill \\ & \le & \sum _{n\ge N^{\prime }}\sum _{v\in {\mathsf{T}}_n}\exp \left(q{S}_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))-(s-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)-ϵq/{\beta }_0)S_n\Phi \left(v\right)-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)S_n\Phi \left(v\right)\right)\hfill \\ & \le & \sum _{n\ge N^{\prime }}\sum _{v\in {\mathsf{T}}_n}\exp \left(q{S}_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))-({\tilde{\mathcal{P}}}_{\alpha }\left(q\right)+\eta )S_n\Phi \left(v\right)\right).\hfill \end{array}$$

Since, from the definition of ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$, we have

$$\begin{array}{ccc}\hfill \mathbb{E}[\sum _{n\ge 1}\sum _{v\in {\mathsf{T}}_n}& & \exp \left(q{S}_n(\mathsf{X}\left(v\right)-\alpha \mathsf{Y}\left(v\right))-({\tilde{\mathcal{P}}}_{\alpha }\left(q\right)+\eta )S_n\Phi \left(v\right)\right)]\hfill \\ & =& \sum _{n\ge 1}\mathbb{E}{\left(\sum _{i=1}^N\exp \left(q({\mathsf{X}}_i-\alpha {\mathsf{Y}}_i)-({\tilde{\mathcal{P}}}_{\alpha }\left(q\right)+\eta )S_n{\Phi }_i\right)\right)}^n<\infty ,\hfill \end{array}$$

it follows that, a.s., ${\overline{\mathcal{P}}}_{\Phi }^s(\left[u\right]\cap F_N)=0$. This holds for all u of a generation larger than N, so ${\mathcal{P}}_{\Phi }^s\left(F_N\right)=0$ a.s. Since this is true for all $N\ge 1$, we obtain a.s. ${\mathcal{P}}_{\Phi }^s\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)=0$ and ${\mathrm{Dim}}_{\Phi }E\left(\alpha \right)\le {\tilde{\mathcal{P}}}_{\alpha }\left(q\right)+ϵq/{\beta }_0+\eta$, for all $q>0,ϵ>0$ and $\eta >0$. Consequently, a.s.

$${\mathrm{Dim}}_{\Phi }E\left(\alpha \right)\le \underset{q\ge 0}{\inf }{\tilde{\mathcal{P}}}_{\alpha }\left(q\right).$$

Similarly, we have, for $q\le 0$, a.s. ${\mathrm{Dim}}_{\Phi }E\left(\alpha \right)\le {\inf }_{q\le 0}{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)-ϵq/{\beta }_0+\eta$, for all $\eta >0$ and $ϵ>0$. Consequently, a.s. ${\mathrm{Dim}}_{\Phi }E\left(\alpha \right)\le {\inf }_{q\le 0}{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$. Finally we have a.s.

$${\mathrm{Dim}}_{\Phi }E\left(\alpha \right)\le \underset{q\in \mathbb{R}}{\inf }{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)={\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right).$$

□

4.2. Lower Bound of Sets ${\mathrm{E}}_{\chi }\left(\alpha \right)$ Under Metric $d_{\Phi }$

Proof of Theorem 4.

First notice, according to Lemma 4, that

$${\mathrm{E}}_{\chi }\left(\alpha \right)=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n(\mathsf{X}-\alpha \mathsf{Y})\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=0\right\}=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n(\mathsf{X}-\alpha \mathsf{Y})\left(\mathsf{t}\right)}{S_n\Phi \left(\mathsf{t}\right)}}=0\right\}.$$

A calculation shows that

$${\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)={\displaystyle \frac{\mathbb{E}\left({\sum }_{i=1}^N\left({\mathsf{X}}_i-\alpha {\mathsf{Y}}_i\right)\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i\right)\right)}{\mathbb{E}\left({\sum }_{i=1}^N{\Phi }_i\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i\right)\right)}}.$$

(30)

The proof of Theorem 4 is based on the mass distribution principle (Theorem 2). Since studying the set ${\mathrm{E}}_{\chi }\left(\alpha \right)$ under a random metric reduces to analyzing the case where the corresponding level set is related to 0, it essentially requires establishing the existence of $q_{\alpha }$ such that

$$\left\{\begin{array}{cc}{\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)=0\hfill & \\ {\displaystyle \underset{n\to \infty }{\lim }\frac{\log {\mu }_{q_{\alpha }}\left(\left[{\mathsf{t}}_{|n}\right]\right)}{\log \left(\right|{\mathsf{t}}_{|n}\left|\right)}={\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right)}\hfill & .\hfill \end{array}\right.$$

The next lemma is a generalization of [25], Proposition 3.2 (it was proven when ${\mathsf{Y}}_i={\Phi }_i=1$ for all $1\le i\le N$).   □

Lemma 5.

Recall the function $\Pi (q,\alpha )$ defined in (6) and assume (8). For each $\alpha \in int\left(I_{\chi }\right)$,

• The function $q⟼\Pi (q,\alpha )$ reaches its infimum at some $q\left(\alpha \right)\in \mathbb{R}$;
• There exists a unique $q=q_{\alpha }$ such that ${\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)=0$.
• The function $\alpha \in int\left(I_{\chi }\right)⟼q_{\alpha }$ is analytic.

Proof. 

Let ${\lambda }_0$ denote the supremum in (8) and fix $q\in \mathbb{R}$.

1.

We define, for $t\in \mathbb{R}$, the function

$$h\left(t\right)=\log \mathbb{E}\left(\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i-t{\mathsf{Y}}_i\right)\right).$$

We have $h\left(0\right)=\tilde{P}\left(q\right)$ and $h^{\prime }\left(0\right)=-\mathbb{E}\left({\sum }_{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i\right)\ge -{\lambda }_0$. Moreover h is convex, so for all $t\ge 0$ we have

$$h\left(t\right)\ge t{h}^{\prime }\left(0\right)+h\left(0\right)\ge h\left(0\right)-{\lambda }_{0}t.$$

In particular, for $t=q\alpha ,$ where $\alpha \in int\left(I_{\chi }\right)$, we have

$$\Pi (q,\alpha )=h\left(q\alpha \right)\ge \tilde{P}\left(q\right)-{\lambda }_{0}q\alpha .$$

Now, it follows from the convexity of $q⟼\Pi (q,\alpha )$, the strict convexity of $\tilde{P}\left(q\right)-{\lambda }_{0}q\alpha$, and the fact that $\tilde{P}\left(q\right)-{\lambda }_{0}q\alpha$ reaches its infimum that the function $q⟼\Pi (q,\alpha )$ reaches its infimum at some $q\left(\alpha \right)\in \mathbb{R}$.

2.

Let $\alpha \in int\left(I_{\chi }\right)$. Then, for $t\in \mathbb{R}$, write

$$\begin{array}{ccc}\hfill \tilde{h}\left(t\right)& =& \log \mathbb{E}\left(\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-t{\Phi }_i\right)\right)\hfill \\ & =& \log \left[\mathbb{E}\left(\sum _{i=1}^N\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-\Pi (q,\alpha )-t{\Phi }_i\right)\right)\right]+\Pi (q,\alpha ).\hfill \end{array}$$

We have $\tilde{h}\left(0\right)=\Pi (q,\alpha )$ and ${\tilde{h}}^{\prime }\left(0\right)=-\mathbb{E}\left({\sum }_{i=1}^N{\Phi }_i\exp \left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-\Pi (q,\alpha )\right)\right)\ge -\lambda$, where $\lambda$ is the supremum in (29). Moreover, $\tilde{h}$ is convex, so for all $t\ge 0$ we have

$$\tilde{h}\left(t\right)\ge t{\tilde{h}}^{\prime }\left(0\right)+\tilde{h}\left(0\right)\ge \tilde{h}\left(0\right)-\lambda t.$$

In particular, for $t={\tilde{\mathcal{P}}}_{\alpha }\left(q\right),$ we have

$$0=\tilde{h}\left({\tilde{\mathcal{P}}}_{\alpha }\left(q\right)\right)\ge \Pi (q,\alpha )-\lambda {\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$$

and then

$${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)\ge {\lambda }^{-1}\Pi (q,\alpha ).$$

(31)

Now it follows from the convexity in q of ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$, the strict convexity of $\Pi (q,\alpha )$, and the fact that $\Pi (q,\alpha )$ reaches its infimum that ${\tilde{\mathcal{P}}}_{\alpha }\left(q\right)$ reaches its infimum at some $q_{\alpha }\in \mathbb{R}$. If there are two distinct $q_{\alpha }$ and $q_{\alpha }^{\prime }$, then ${\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)=0$ over $[q_{\alpha },q_{\alpha }^{\prime }]$, i.e., due to (30) below,

$$g\left(q\right)=\mathbb{E}\left(\sum _{i=1}^N({\mathsf{X}}_i-\alpha {\mathsf{Y}}_i)\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)=0$$

and then

$$g^{\prime }\left(q\right)=\mathbb{E}\left(\sum _{i=1}^N{({\mathsf{X}}_i-\alpha {\mathsf{Y}}_i)}^2\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)=0$$

over $[q_{\alpha },q_{\alpha }^{\prime }]$; hence, ${\mathsf{X}}_i=\alpha {\mathsf{Y}}_i$ a.s. for all $1\le i\le N$. But, this contradicts (9).

3.

An examination of the derivative of the function $q⟼{\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)$ at $q_{\alpha }$ shows that it is invertible, except if ${\mathsf{X}}_i-\alpha {\mathsf{Y}}_i=0$ for all $1\le i\le N$ a.s., which is forbidden by condition (9). Then, the invertibility of $q⟼{\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)$ at $q_{\alpha }$ for each $\alpha \in int\left(I_{\chi }\right)$ makes it possible to apply the implicit function theorem to $(\alpha ,q)⟼{\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)$ at $(\alpha ,q_{\alpha })$, and then the function $\alpha ⟼q_{\alpha }$ is analytic.

□

Remark 6.

Let $\alpha \in int\left(I_{\chi }\right)$, then ${\displaystyle \underset{q\in \mathbb{R}}{\inf }\Pi (q,\alpha )>0.}$ It follows from (31) that ${\displaystyle {\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right)=\underset{q\in \mathbb{R}}{\inf }{\tilde{\mathcal{P}}}_{\alpha }\left(q\right)>0.}$

For each $(q,\alpha )\in {\mathbb{R}}^2$, a Mandelbrot measure ${\mu }_{q,\alpha }$ on $\partial \mathsf{T}$ is associated with the vectors $(N_u,\exp (q({\mathsf{X}}_{u1}-\alpha {\mathsf{Y}}_{u1})-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_{u1}),\exp (q({\mathsf{X}}_{u2}-\alpha {\mathsf{Y}}_{u2})-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_{u2}),\dots )$, $u\in {\bigcup }_{n\ge 0}{\mathbb{N}}_{+}^n$. Moreover, the measure ${\mu }_{q,\alpha }$ is non-degenerate if and only if

$$-\mathbb{E}\left(\sum _{i=1}^N\left(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i\right)\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)>0.$$

Since $\mathbb{E}\left({\sum }_{i=1}^N{\Phi }_i\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)>0$, we have that ${\mu }_{q,\alpha }$ is non-degenerate if and only if

$${\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)\right)={\tilde{\mathcal{P}}}_{\alpha }\left(q\right)-q{\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q\right)>0.$$

(32)

In addition (see Section 2.2), with a probability of 1, we have ${\mu }_{q,\alpha }$ a.e.

$$\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{n}}={\alpha }_{\mathsf{X}}(q,\alpha )\mathrm{and}\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}={\alpha }_{\mathsf{Y}}(q,\alpha ),$$

(33)

where

$${\alpha }_{\mathsf{X}}(q,\alpha )=\mathbb{E}\left(\sum _{i=1}^N{\mathsf{X}}_i\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)$$

$${\alpha }_{\mathsf{Y}}(q,\alpha )=\mathbb{E}\left(\sum _{i=1}^N{\mathsf{Y}}_i\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right).$$

Let $\alpha \in int\left(I_{\chi }\right)$ and construct the Mandelbrot measure ${\mu }_{q_{\alpha },\alpha }$ on $\partial \mathsf{T}$, where $q_{\alpha }$ is defined in Lemma 5 such that ${\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)\right)>0$ and ${\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)=0$; that is, using (30) and (33),

$${\alpha }_{\mathsf{X}}(q_{\alpha },\alpha )=\alpha {\alpha }_{\mathsf{Y}}(q_{\alpha },\alpha ).$$

The fact that ${\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)\right)={\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right)>0$ implies that this measure is a.s. positive by (32). Moreover, with a probability of 1, we have

$$\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}={\displaystyle \frac{{\alpha }_{\mathsf{X}}(q_{\alpha },\alpha )}{{\alpha }_{\mathsf{Y}}(q_{\alpha },\alpha )}}=\alpha {\mu }_{q_{\alpha },\alpha }\text{-}\mathrm{a}.\mathrm{e}..$$

(34)

It follows that ${\mu }_{q_{\alpha },\alpha }\left({\mathrm{E}}_{\chi }\left(\alpha \right)\right)=1$. In addition, using Theorem 1, we have a.s. for ${\mu }_{q_{\alpha },\alpha }$-almost every t,

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{\lim }{\displaystyle \frac{\log {\mu }_{q_{\alpha }}\left(\left[t_{|n}\right]\right)}{\log \left(\right|t_{|n}\left|\right)}}& =& \underset{n\to \infty }{\lim }{\displaystyle \frac{\log {\mu }^{\prime }\left(\left[t_{|n}\right]\right)}{-S_n\Phi \left(\mathsf{t}\right)}}\hfill \\ & =& -\mathbb{E}\left(\sum _{i=1}^N(q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right)/{\alpha }_{\Phi }(q_{\alpha },\alpha )\hfill \\ & =& {\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left({\tilde{\mathcal{P}}}_{\alpha }^{\prime }\left(q_{\alpha }\right)\right)={\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right),\hfill \end{array}$$

where ${\alpha }_{\Phi }(q,\alpha )=\mathbb{E}\left({\sum }_{i=1}^N{\Phi }_i\exp (q{\mathsf{X}}_i-q\alpha {\mathsf{Y}}_i-{\tilde{\mathcal{P}}}_{\alpha }\left(q\right){\Phi }_i)\right).$ Therefore, since $\partial \mathsf{T}$ is endowed with the random metric $d_{\Phi }$, the Mandelbrot measure ${\mu }_{q_{\alpha },\alpha }$ is exact dimensional with dimension ${\tilde{\mathcal{P}}}_{\alpha }^{\ast }\left(0\right).$ Finally, we deduce the result by the mass distribution principle (Theorem 2).

5. Conclusions

Our results provide a precise description of the multifractal structure of level sets associated with branching random walks on a supercritical Galton–Watson tree. We computed the Hausdorff and packing dimensions of the set ${\mathrm{E}}_{\chi }(\alpha ,\beta )$, which develops a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting. The present study opens several avenues for further research:

1.

Vector-valued branching random walks can be considered, and the Hausdorff and packing dimensions of the set $\mathrm{E}(\alpha ,\beta )$ can be computed. The vectorial case of the set $\mathrm{E}\left(\alpha \right)$ was considered in [23].

2.

The speed of convergence of these random branching walks can be studied under the random metric $d_{\Phi }$. More precisely, given a sequence $s≔\left\{s_n\right\}$, we consider the set

$${\mathrm{E}}_{\chi }^s(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:\underset{n\to \infty }{\lim }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}{S}_n\mathsf{Y}\left(\mathsf{t}\right)-n\beta \sim s_n\right\}.$$

Obviously, if $s_n=o\left(n\right)$, then ${\mathrm{E}}_{\chi }^s(\alpha ,\beta )\subset {\mathrm{E}}_{\chi }(\alpha ,\beta )$, and it is woudl be interesting to compute a sufficient condition on s so that

$$\dim {\mathrm{E}}_{\chi }^s(\alpha ,\beta )=\dim {\mathrm{E}}_{\chi }(\alpha ,\beta ).$$

3.

Finally, this framework may also be connected to applications in physics, biology, and network dynamics, where Galton–Watson trees and branching processes naturally appear, providing a tool for multifractal analysis to quantify heterogeneity in complex empirical systems.