Laws of the k-Iterated Logarithm of Weighted Sums in a Sub-Linear Expected Space

Abstract

The law of the iterated logarithm precisely refines the law of large numbers and plays a fundamental role in probability limit theory. The framework of sub-linear expectation spaces substantially extends the classical concept of probability spaces. In this study, we employ a methodology that differs from the traditional probabilistic approach to study the k-iterated logarithm law for weighted sums of stable random variables with the exponent $\alpha \in (0,2)$ within sub-linear expectation space, establishing a highly general form of the k-iterated logarithm law in this context. The obtained results include Chover’s law of the iterated logarithm, as well as the laws for partial sums and moving average processes, thereby extending many corresponding results obtained in classical probability spaces.

1. Introduction

Statistical modeling is a cornerstone of both theoretical and applied statistical sciences, with estimation playing an essential role in the process. A central concern in estimation involves asymptotic behavior, which is commonly studied through limit theory. Within classical probability theory, statistical models typically rely on the assumption that errors—and consequently, response variables—follow a uniquely defined probability distribution. This assumption of distributional determinacy underpins a well-developed body of theories and methods in traditional statistical inference.

However, in many modern applications such as economics and finance, data frequently exhibit complex uncertainty that cannot be captured by deterministic distributional assumptions. The inherent variability and ambiguity in such data violate the core premises of classical modeling strategies. This renders conventional probabilistic approaches inadequate, creating a persistent methodological gap in handling probabilistic and distributional uncertainties within statistical modeling.

To address these challenges, Peng (2006 [1], 2008 [2]) introduced the notion of sub-linear expectation spaces, which extend the classical probabilistic framework through a generalized expectation-based approach. This innovation offers a promising foundation for statistical reasoning under uncertainty. However, the non-additive nature of sub-linear expectations and the concept of solubility render conventional limit-theoretic tools ineffective, creating a significant obstacle in extending classical limit theorems—particularly the laws of the iterated logarithm—to this broader framework.

Motivated by this theoretical gap, our work aims establish Chover’s law of the iterated logarithm for weighted sums of independent and identically distributed random variables under sub-linear expectations. Through this extension, we seek to enhance the mathematical foundation for statistical modeling in environments characterized by distributional uncertainty and provide new tools for asymptotic analysis beyond traditional probability spaces.

First, we introduce Chover’s law of the iterated logarithm of probability space. Let $\{X,X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables of symmetric, stable distribution with the exponent $\alpha ,0<\alpha <2$, i.e.,

$$P\left(\right|X|>x)={\displaystyle \frac{c\left(x\right)l\left(x\right)}{x^{\alpha }}}\mathrm{for}x>0,$$

where $c\left(x\right)\ge 0,\underset{x\to \infty }{lim}c\left(x\right)=c>0$ and $l\left(x\right)\ge 0$ is a slowly varying function.

Chover (1966 [3]) established the following:

$$\underset{n\to \infty }{lim\; sup}{\left|{\displaystyle \frac{S_n}{n^{1/\alpha }}}\right|}^{{\displaystyle \frac{1}{\log \log n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}..$$

The above equation is called Chover’s law of the iterated logarithm (CLIL). Since its establishment, CLIL has inspired many probabilistic statisticians to study and obtain various versions of it. Wichura (1974 [4]), Vasudeva (1984 [5]), and Qi and Cheng (1996 [6]) obtained CLIL for partial sums of asymmetric, stable distribution. Lu and Qi (2006 [7]) and Wang (2014 [8]) obtained CLILs for trimmed sums and operator-stable Lévy processes, respectively. Some CLILs for weighted sums were obtained by Chen (2002 [9]), Peng and Qi (2003 [10]), Chen and Qi (2006 [11]), and Trapani (2014 [12]). Furthermore, Peng and Qi (2003 [10]), Chen and Qi (2006 [11]), and Wu and Jiang (2010 [13]) obtained Chover’s laws of the k-iterated logarithm (k-CLIL).

Peng (2006 [1], 2008 [2]) noted that in the past two decades, limit theory of sub-linear expected spaces has developed rapidly. Peng (2008 [2]) obtained the central limit theorem, and Zhang (2016 [14], 2016 [15]) established some inequalities about partial sums, including exponential inequalities, Rosenthal’s inequalities, Kolmogorov’s strong law of larger numbers, and the Hartman–Wintner law of the iterated logarithm. Hu (2016 [16]) and Wu and Jiang (2018 [17]) further studied the strong law of larger numbers in a general form, Wu and Jiang (2018 [17]), Wu and Lu (2020 [18]), Wu (2023 [19]) studied CLIL, etc. Recently, Tartakovsky (2023 [20]) generalized complete convergence and r-quick convergence in the law of large numbers and applied these convergence modes to solve challenging statistical problems in hypothesis testing and changepoint detection for non-i.i.d. models, providing new insights for the application of limit theory in sub-linear expectation spaces. Sun et al. (2023 [21]) investigated complete convergence and p-th moment convergence for weighted sums of negative-dependence (ND) random variables under sub-linear expectation space and established equivalent conditions using moment inequality and truncation methods, enriching limit theory of dependent random variables in a sub-linear expectation framework. Guo and Meng (2023 [22]) studied the Marcinkiewicz–Zygmund-type strong law of large numbers under sub-linear expectations.

Wu and Jiang (2018 [17]) studied and extended Chover’s laws of the k-iterated logarithm (k-CLIL) from sequences of the probability space to sequences of sub-linear expectation. In this study, based on the work of Wu and Jiang (2018 [17]), we further determine that the k-CLIL for weighted sums also holds in sub-linear expected space. This article promotes and expands the corresponding results of Wu and Jiang (2018 [17]).

The basic structure of this article is as follows. In Section 2, for the benefit of the readers, we introduce the basic concepts and properties of sub-linear expected space. In Section 3, we propose three lemmas, among which Lemmas 2 and 3 are important conclusions regarding almost sure convergence theorems of partial sums and weighted sums in the sub-linear expected space, respectively, which play a key role in proving the main results. The k-CLIL for weighted sums under sub-linear expectation is established in Section 4. In Section 5, we provide two examples to prove that the limits in Theorem 2 have various forms.

2. The Concept and Properties of Sub-Linear Expected Space

The general framework of the sub-linear expectation in a general function space was introduced by Peng (2006 [1], 2008 [2]). Let $(\mathrm{\Omega },\mathcal{F})$ be a given measurable space, and let $\mathcal{H}$ be a linear space of real functions defined on $(\mathrm{\Omega },\mathcal{F})$, such that if $X_1,\dots ,X_n\in \mathcal{H}$, then $\phi (X_1,\dots ,X_n)\in \mathcal{H}$ for each $\phi \in C_{l,Lip}\left({\mathbb{R}}_n\right)$, where $C_{l,Lip}\left({\mathbb{R}}_n\right)$ denotes the linear space of (local Lipschitz) functions $\phi$ satisfying

$$|\phi \left(\mathbf{x}\right)-\phi \left(\mathbf{y}\right)|\le {c(1+|\mathbf{x}|}^m+{\left|\mathbf{y}\right|}^m\left)\right|\mathbf{x}-\mathbf{y}|,\forall \mathbf{x},\mathbf{y}\in {\mathbb{R}}_n,$$

for some $c>0,m\in \mathbb{N}$ depending on $\phi$. $\mathcal{H}$ is considered a space of random variables. In this case, we denote $X\in \mathcal{H}$.

Definition 1.

A function $\widehat{\mathbb{E}}:\mathcal{H}\to [-\infty ,\infty ]$ is said to be a sub-linear expectation if it satisfies all $X,Y\in \mathcal{H}$,

(a) 

Monotonicity: If $X\ge Y$ then $\widehat{\mathbb{E}}X\ge \widehat{\mathbb{E}}Y$;

(b) 

Constant preservation: $\widehat{\mathbb{E}}c=c$;

(c) 

Sub-additivity: $\widehat{\mathbb{E}}(X+Y)\le \widehat{\mathbb{E}}X+\widehat{\mathbb{E}}Y$ whenever $\widehat{\mathbb{E}}X+\widehat{\mathbb{E}}Y$ is not of the form $+\infty -\infty$ or $-\infty +\infty$;

(d) 

Positive homogeneity: $\widehat{\mathbb{E}}\left(\lambda X\right)=\lambda \widehat{\mathbb{E}}X,\lambda \ge 0$.

The triple $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$ is called a sub-linear expectation space. The conjugate expectation $\widehat{\epsilon }$ of $\widehat{\mathbb{E}}$ is defined by

$$\widehat{\epsilon }X:=-\widehat{\mathbb{E}}(-X),\forall X\in \mathcal{H}.$$

From the definition, it is easily shown that for all $X,Y\in \mathcal{H}$,

$$\widehat{\epsilon }X\le \widehat{\mathbb{E}}X,\widehat{\mathbb{E}}(X+c)=\widehat{\mathbb{E}}X+c,|\widehat{\mathbb{E}}(X-Y)|\le \widehat{\mathbb{E}}|X-Y|\mathrm{and}\widehat{\mathbb{E}}(X-Y)\ge \widehat{\mathbb{E}}X-\widehat{\mathbb{E}}Y$$

If $\widehat{\mathbb{E}}Y=\widehat{\epsilon }Y$, then $\widehat{\mathbb{E}}(X+aY)=\widehat{\mathbb{E}}X+a\widehat{\mathbb{E}}Y$ for any $a\in \mathbb{R}$. Sun et al. (2023 [21]) further extended the application of sub-linear expectation to ND random variables and pointed out that the properties of independent sequences can be generalized to ND sequences under sub-linear expectation, which is of great significance for studying the limit behavior of dependent random variables in uncertain environments.

Definition 2.

A function $V:\mathcal{F}\to [0,1]$ is called a capacity if

$$V(\varnothing )=0,V\left(\mathrm{\Omega }\right)=1\mathrm{and}V\left(A\right)\le V\left(B\right)\mathrm{for}\forall A\subseteq B,A,B\in \mathcal{F}.$$

It is sub-additive if $V(A\bigcup B)\le V\left(A\right)+V\left(B\right)$ for all $A,B\in \mathcal{F}$. In the sub-linear space $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$, we denote a pair $(\mathbb{V},\mathbf{\nu })$ of capacities by

$$\mathbb{V}\left(A\right):=inf\{\widehat{\mathbb{E}}\xi ;I\left(A\right)\le \xi ,\xi \in \mathcal{H}\},\mathbf{\nu }\left(A\right):=1-\mathbb{V}\left(A^c\right),\forall A\in \mathcal{F},$$

where $A^c$ is the complement set of A.

By definition of $\mathbb{V}$ and $\mathbf{\nu }$, it is obvious that $\mathbb{V}$ is sub-additive, and

$$\mathbf{\nu }\left(A\right)\le \mathbb{V}\left(A\right),\forall A\in \mathcal{F},$$

$$\widehat{\mathbb{E}}f\le \mathbb{V}\left(A\right)\le \widehat{\mathbb{E}}g,\widehat{\epsilon }f\le \mathbf{\nu }\left(A\right)\le \widehat{\epsilon }g,\mathrm{if}f\le I\left(A\right)\le g,f,g\in \mathcal{H}.$$

Definition 3.

(i) A sub-linear expectation $\widehat{\mathbb{E}}:\mathcal{H}\to \mathbb{R}$ is countably sub-additive if it satisfies

$$\widehat{\mathbb{E}}\left(X\right)\le {\displaystyle \sum _{n=1}^{\infty }}\widehat{\mathbb{E}}\left(X_n\right),\mathrm{whenever}X\le {\displaystyle \sum _{n=1}^{\infty }}{X}_n,X,X_n\in \mathcal{H},X\ge 0,X_n\ge 0,n\ge 1.$$

It is continuous if it satisfies

$$\widehat{\mathbb{E}}\left(X_n\right)\uparrow \widehat{\mathbb{E}}\left(X\right),\mathrm{if}0\le X_n\uparrow X,\mathrm{and}\widehat{\mathbb{E}}\left(X_n\right)\downarrow \widehat{\mathbb{E}}\left(X\right),\mathrm{if}0\le X_n\downarrow X,\mathrm{where}X,X_n\in \mathcal{H}.$$

(ii) A function $V:\mathcal{F}\to [0,1]$ is called to be countably sub-additive if

$$V\left({\displaystyle \bigcup _{n=1}^{\infty }}{A}_n\right)\le {\displaystyle \sum _{n=1}^{\infty }}V\left(A_n\right),\forall A_n\in \mathcal{F}.$$

It is continuous if it satisfies

$$V\left(A_n\right)\uparrow V\left(A\right),\mathrm{if}{A}_n\uparrow A,\mathrm{and}V\left(A_n\right)\downarrow V\left(A\right),\mathrm{if}{A}_n\downarrow A,\mathrm{where}A,A_n\in \mathcal{F}.$$

It is obvious that a continuous sub-additive capacity V (a sub-linear expectation $\widehat{\mathbb{E}}$) is countably sub-additive.

Definition 4

(Peng (2006 [1], 2008 [2]), Zhang (2016a [14])).

(i) 

(Identical distribution) Let $X_1$ and $X_2$ be two random variables defined in the sub-linear expectation spaces $({\mathrm{\Omega }}_1,{\mathcal{H}}_1,{\widehat{\mathbb{E}}}_1)$ and $({\mathrm{\Omega }}_2,{\mathcal{H}}_2,{\widehat{\mathbb{E}}}_2)$, respectively. They are identically distributed if

$${\widehat{\mathbb{E}}}_1\left(\phi \left(X_1\right)\right)={\widehat{\mathbb{E}}}_2\left(\phi \left(X_2\right)\right),\forall \phi \in C_{l,Lip}\left(\mathbb{R}\right),$$

whenever the sub-expectations are finite. A sequence $\{X_n;n\ge 1\}$ of random variables is said to be identically distributed if for each $i\ge 1$, $X_i$ and $X_1$ are identically distributed.

(ii) 

(Independence) In a sub-linear expectation space $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$, a random vector $\mathbf{Y}=(Y_1,\dots ,Y_n)$, $Y_i\in \mathcal{H}$ is said to be independent of another random vector $\mathbf{X}=(X_1,\dots ,X_m),$$X_i\in \mathcal{H}$ under $\widehat{\mathbb{E}}$ if for each test function $\phi \in C_{l,Lip}({\mathbb{R}}_m\times {\mathbb{R}}_n)$, we have $\widehat{\mathbb{E}}\left(\phi (\mathbf{X},\mathbf{Y})\right)=\widehat{\mathbb{E}}\left[\widehat{\mathbb{E}}\left(\phi (\mathbf{x},\mathbf{Y})\right){|}_{\mathbf{x}=\mathbf{X}}\right]$ whenever $\overline{\phi }\left(\mathbf{x}\right):=\widehat{\mathbb{E}}\left(\left|\phi \right(\mathbf{x},\mathbf{Y}\left)\right|\right)<\infty$ for all $\mathbf{x}$ and $\widehat{\mathbb{E}}\left(|\overline{\phi }\left(\mathbf{X}\right)|\right)<\infty$.

(iii) 

(Independent random variables) A sequence of random variables $\{X_n;n\ge 1\}$ is said to be independent if $X_{i+1}$ is independent of $(X_1,\dots ,X_i)$ for each $i\ge 1$.

(iv) 

(Extended independence) A sequence of random variables $\{X_n;n\ge 1\}$ is said to be extended independent if

$$\widehat{\mathbb{E}}\left({\displaystyle \prod _{i=1}^n}{\phi }_i\left(X_i\right)\right)={\displaystyle \prod _{i=1}^n}\widehat{\mathbb{E}}\left({\phi }_i\left(X_i\right)\right),\forall n\ge 2,\forall 0\le {\phi }_i\left(x\right)\in C_{l,Lip}\left(\mathbb{R}\right).$$

It can be shown that the independence implies extended independence, and if $\{X_n;n\ge 1\}$ is a sequence of extended independent random variables and $f_1\left(x\right),f_2\left(x\right),\dots \in C_{l,Lip}\left(\mathbb{R}\right)$, then $\{f_n\left(X_n\right);n\ge 1\}$ is also a sequence of extended independent random variables. Zhang (2022 [23]) further systematically studied the basic properties of extended independent and extended negatively dependent random variables under sub-linear expectations, including their moment inequalities and convergence behaviors, providing a more comprehensive theoretical basis for the application of extended independent random variables in limit theory (such as the proof of almost sure convergence in Lemmas 2 and 3).

In the following, let $\{X_n;n\ge 1\}$ be a sequence of random variables in a sub-linear expectation space $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$, and $S_n={\sum }_{i=1}^n{X}_i$. The symbol c stands for a generic positive constant, which may differ from one place to another.

3. Main Lemmas and Their Proofs

To prove our results, we need the following three lemmas.

Lemma 1

(Qi and Cheng 1996 [6]). Suppose that $h\left(x\right)$ is a slowly varying function at infinity and $g\left(x\right)$ is a positive function with $\underset{x\to \infty }{lim}g\left(x\right)=\infty$. Then, for any given $\delta >0$, there exists an $x_0>0$ such that

$$g^{-\delta }\left(x\right)\le \underset{x\le y\le xg\left(x\right)}{inf}{\displaystyle \frac{h\left(y\right)}{h\left(x\right)}}\le \underset{x\le y\le xg\left(x\right)}{sup}{\displaystyle \frac{h\left(y\right)}{h\left(x\right)}}\le g^{\delta }\left(x\right)\mathrm{for}\mathrm{all}x>x_0.$$

In the sub-linear expectation space, the almost sure convergence of random variable sequences differs from the traditional probability space. We first define the concept of almost sure in the sub-linear expected space.

Definition 5.

For an arbitrary event $A\in \mathcal{F}$, A is said to be almost surely V (denoted by A a.s. V) if $V\left(A^c\right)=0$, where $A^c$ is the complement set of A.

In particular, a sequence of random variables $\{X_n;n\ge 1\}$ is said to converge to X almost surely V, denoted by $X_n\to X$ a.s. V, as $n\to \infty$ if $V(X_n\nrightarrow X)=0$.

V can be replaced by $\mathbb{V}$ and $\mathbf{\nu }$. By $\mathbf{\nu }\left(A\right)\le \mathbb{V}\left(A\right)$ and $\mathbf{\nu }\left(A\right)+\mathbb{V}\left(A^c\right)=1$ for any $A\in \mathcal{F}$, it is obvious that $X_n\to X$ a.s. $\mathbb{V}$ implies $X_n\to X$ a.s. $\mathbf{\nu }$. However, we must point out that $X_n\to X$ a.s. $\mathbf{\nu }$ does not imply $X_n\to X$ a.s. $\mathbb{V}$. Wu and Lu (2020 [18]) gave a counter-example of this, as follows.

Example 1

(Wu and Lu, Example 3.3 (2020 [18])). Let $X_n$ be independent G-normal random variables with $X_n\sim \mathcal{N}(0,[1/4^{2n},1])$ in a sub-linear expectation space $(\mathrm{\Omega },\mathcal{H},\widehat{\mathbb{E}})$. $\widehat{\mathbb{E}}$ and $\mathbb{V}$ are continuous. Then, $X_n\to 0$ a.s. ν but not $X_n\to 0$ a.s.$\mathbb{V}$.

Therefore, in sub-linear expectation spaces, sure convergence is essentially different from the ordinary probability space, and studying it is much more complex and difficult.

To ensure that the sequence of truncated random variables also has extended independence, we need to modify the indicator function by the functions in $C_{l,Lip}$.

For $0<\mu <1$, let $g\left(x\right)\in C_{l,Lip}\left(\mathbb{R}\right)$ be a non-increasing function such that $0\le g\left(x\right)\le 1$ for all x and $g\left(x\right)=1$ if $x\le \mu$, $g\left(x\right)=0$ if $x>1$. For any $i\le n$, let

$$X_i\left(a_n\right)=X_{i}g\left({\displaystyle \frac{|X_i|}{a_n}}\right).$$

(1)

The following two lemmas are criteria for the almost sure convergence of partial sums and weighted sums of sub-linear expectation spaces. These two lemmas are almost sure convergence theorems of partial sums and weighted sums in sub-linear expected space, respectively, which play a key role in proving the main results of this study.

Lemma 2

(Wu and Jiang 2018 [17], Theorem 3.3). Assume that $\{X_n;n\ge 1\}$ is a sequence of extended independent and identically distributed (e.i.i.d.) random variables, $\mathbb{V}$ is continuous, and $\{a_n;n\ge 1\}$ is a sequence of positive numbers with $a_n/n^{1/\beta }\uparrow$ for some $\beta \in (0,2)$ and $\underset{n\ge 1}{sup}{a}_{2n}/a_n<\infty$.

(i) If

$${\displaystyle \sum _{n=1}^{\infty }}\mathbb{V}\left(\right|X_1|>a_n)<\infty ,$$

(2)

then

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{S_n-c_n}{a_n}}\le 0\mathrm{a}.\mathrm{s}.\mathbb{V},$$

(3)

where $c_n=n\widehat{\mathbb{E}}\left(X_1\left(a_n\right)\right)$ and $X_1\left(a_n\right)$ defined by (1), in particular, $c_n=0$ if ${\displaystyle \frac{a_n}{n}}\uparrow \infty$ and $c_n=n\widehat{\mathbb{E}}{X}_1$ if ${\displaystyle \frac{n}{a_n}}\uparrow ,\widehat{\mathbb{E}}\left|X_1\right|<\infty ,\widehat{\mathbb{E}}$ is countably sub-additive. And

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{S_n-{\tilde{c}}_n}{a_n}}\ge 0\mathrm{a}.\mathrm{s}.\mathbb{V},$$

(4)

where ${\tilde{c}}_n=n\widehat{\epsilon }\left(X_1\left(a_n\right)\right)$, in particular, ${\tilde{c}}_n=0$ if ${\displaystyle \frac{a_n}{n}}\uparrow \infty$ and ${\tilde{c}}_n=n\widehat{\epsilon }{X}_1$ if ${\displaystyle \frac{n}{a_n}}\uparrow ,\widehat{\mathbb{E}}\left|X_1\right|<\infty ,\widehat{\mathbb{E}}$ is countably sub-additive.

(ii) If

$${\displaystyle \sum _{n=1}^{\infty }}\mathbb{V}\left(\right|X_1|>a_n)=\infty ,$$

(5)

then

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|S_n-d_n|}{a_n}}=\infty \mathrm{a}.\mathrm{s}.\mathbf{\nu }$$

for every sequence $\{d_n;n\ge 1\}$ such that

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|d_n-d_{n-1}|}{a_n}}=c<\infty .$$

(6)

Lemma 3.

Under the conditions of Lemma 2, one of the following two conditions holds.

(C1) ${\displaystyle \frac{a_n}{n}}\uparrow \infty ,$

(C2) ${\displaystyle \frac{n}{a_n}}\uparrow ,\widehat{\mathbb{E}}\left|X_1\right|<\infty ,\widehat{\mathbb{E}}{X}_1=\widehat{\epsilon }{X}_1,$ and $\widehat{\mathbb{E}}$ is countably sub-additive.

Let $\{a_{n,k};1\le k\le n,n\ge 1\}$ be an array of weights satisfying

$$\begin{array}{c}\hfill \underset{n\ge 1}{sup}\left({\displaystyle \sum _{k=1}^{n-1}}|a_{n,k}-a_{n,k+1}|+|a_{n,n}|\right)<\infty ,\mathrm{and}a:=\underset{n\to \infty }{lim\; inf}\left|a_{n,n}\right|>0.\end{array}$$

(7)

(i) If (2) holds, then

$$\underset{n\to \infty }{lim}{\displaystyle \frac{{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)}{a_n}}=0\mathrm{a}.\mathrm{s}.\mathbb{V},$$

(8)

where $b=0$ if (C1) holds, and $b=\widehat{\mathbb{E}}{X}_1$ if (C2) holds.

(ii) If (5) holds, then

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}=\infty \mathrm{a}.\mathrm{s}.\mathbf{\nu }$$

(9)

for every sequence $\{d_n;n\ge 1\}$ satisfying (6).

Remark 1.

Lemmas 2 and 3 themselves are of great significance. They are powerful tools to study the almost sure convergence of partial sums and weighted sums in sub-linear expectation spaces, respectively. Guo and Meng (2023 [22]) also used similar lemma tools (such as moment inequality and truncation method) to study the Marcinkiewicz–Zygmund-type strong law of large numbers under sub-linear expectations and verified the effectiveness of these tools in solving the limit problem of random variables with general normalizing sequences. Tartakovsky (2023 [20]) emphasized that strengthened r-complete and r-quick convergence in the law of large numbers are crucial for establishing the asymptotic optimality of sequential tests and changepoint detection procedures, which also reflects the practical value of the almost sure convergence results in Lemmas 2 and 3.

Proof of Lemma 3.

If (2) holds, from (3) and (4) in Lemma 2, we obtain

$$\underset{n\to \infty }{lim}{\displaystyle \frac{S_n-C_n}{a_n}}=0\mathrm{a}.\mathrm{s}.\mathbb{V},$$

(10)

where $C_n=0$ if (C1) holds, and $C_n=n\widehat{\mathbb{E}}{X}_1$ if (C2) holds. Noting the fact that $a_n>0$ is a non-decreasing function of n, we can observe that (10) implies

$$\underset{n\to \infty }{lim}{\displaystyle \frac{\underset{1\le k\le n}{max}|S_k-C_k|}{a_n}}=0\mathrm{a}.\mathrm{s}.\mathbb{V},$$

(11)

Set $S_0=0$, by (7)

$$\begin{array}{ccc}\hfill \left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|& =& \left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}((S_k-S_{k-1})-(C_k-C_{k-1}))\right|\hfill \\ \hfill & =& \left|{\displaystyle \sum _{k=1}^{n-1}}(a_{n,k}-a_{n,k+1})(S_k-C_k)+a_{n,n}(S_n-C_n)\right|\hfill \\ & \le & c\underset{1\le k\le n}{max}|S_k-C_k|.\hfill \end{array}$$

This and (11) imply (8).

(ii) If (5) holds, then by P264 in Wu and Jiang (2018 [17]), we obtain

$$\mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|X_n|}{a_n}}>M\right)=1,\mathrm{for}\mathrm{any}M>0.$$

From the combination condition (7), i.e., $\underset{n\to \infty }{lim\; inf}\left|a_{n,n}\right|=a>0$, it follows that

$$\mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|a_{n,n}{X}_n|}{a_n}}>M\right)\ge \mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|X_n|}{a_n}}>M/a\right)=1.$$

(12)

Hence, for any $\{d_n;n\ge 1\}$ that satisfies (6),

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|a_{n,n}{X}_n|}{a_n}}& =\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|\left({\displaystyle {\displaystyle \sum _{k=1}^n}}{a}_{n,k}{X}_k-d_n\right)-\left({\displaystyle {\displaystyle \sum _{k=1}^{n-1}}}{a}_{n,k}{X}_k-d_{n-1}\right)+(d_n-d_{n-1})\right|}{a_n}}\hfill \\ \hfill & \le 2\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle {\displaystyle \sum _{k=1}^n}}{a}_{n,k}{X}_k-d_n\right|}{a_n}}+\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|d_n-d_{n-1}|}{a_n}}\hfill \\ \hfill & =2\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle {\displaystyle \sum _{k=1}^n}}{a}_{n,k}{X}_k-d_n\right|}{a_n}}+c.\hfill \end{array}$$

From the arbitrariness of M, this and (12) yield

$$\mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}>M_1\right)=1,\forall M_1>0.$$

Therefore, from the continuity of $\mathbb{V}$,

$$\begin{array}{cc}\hfill \mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}=\infty \right)& =\mathbb{V}\left(\forall m\ge 1,\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}>m\right)\hfill \\ \hfill & =\mathbb{V}\left({\displaystyle \bigcap _{m=1}^{\infty }}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}>m\right)\right)\hfill \\ \hfill & =\underset{m\to \infty }{lim}\mathbb{V}\left(\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}>m\right)\hfill \\ \hfill & =1.\hfill \end{array}$$

That is

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}{X}_k-d_n\right|}{a_n}}=\infty \mathrm{a}.\mathrm{s}.\mathbf{\nu }$$

Thus, (9) is proved. □

4. Chover’s Law of the k-Iterated Logarithm

Referring to the definition of stable distribution for an exponent in (0, 2) in probability space, we define the stable distribution for an exponent in (0, 2) in sub-linear expectation space as follows.

Definition 6.

A sequence of identically distributed random variables defined in a sub-linear expected space $\{X_n;n\ge 1\}$ is a stable distribution with exponent $\alpha \in (0,2)$ if

$$\begin{array}{c}\hfill \mathbb{V}\left(\right|X_1|>x)={\displaystyle \frac{c\left(x\right)l\left(x\right)}{x^{\alpha }}}\mathrm{for}x>0,\end{array}$$

(13)

where $c\left(x\right)\ge 0,\underset{x\to \infty }{lim}c\left(x\right)=c>0$, and $l\left(x\right)\ge 0$ is a slowly varying function.

From Seneta (1976 [24]), $l\left(x\right)$ is a slowly varying function if and only if

$$l\left(x\right)=c_1\left(x\right)exp\left\{{\int }_1^x{\displaystyle \frac{f\left(u\right)}{u}}\mathrm{d}u\right\},x>0,$$

where $c_1\left(x\right)\ge 0$, $\underset{x\to \infty }{lim}{c}_1\left(x\right)=c_1>0,$ and $\underset{x\to \infty }{lim}f\left(x\right)=0.$

Setting ${lg}_{0}x=x$ and denoting ${lg}_{j}x=ln\{max(\mathrm{e},{lg}_{j-1}x)\}$ for $j\ge 1$, $lgx={lg}_{1}x$, we also assume $k\ge 0$ is a fixed integer. For convenience, the product ${\prod }_{j=1}^k(·)$ is defined as 1 if $k<1$. $A\sim B$ denotes $A/B\to 1$.

Wu and Jiang (2018 [17]) recently proved that, under sub-linear expectation, for a sequence of e.i.i.d. random variables satisfying (13), k-CLIL is as follows: there exist some constants $A_n\in \mathbb{R}$, $B_n>0$ such that

$$\underset{n\to \infty }{lim\; sup}{\left|{\displaystyle \frac{S_n-A_n}{B_n}}\right|}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

(14)

In this study, we aim to demonstrate that the law of the iterated logarithm (14) also holds under sub-linear expectation spaces for generalized weighted sums, including partial sums, moving sums, and other forms of weighted sums.

Set $G\left(x\right)=\mathbb{V}\left(\right|X_1|\ge x)$ and define

$$B\left(x\right)=inf\{y;G(y)\le 1/x\}\mathrm{for}x>0.$$

From (13), $G\left(x\right)$ is a regularly varying function with index $-\alpha$ at infinity. Hence, from De Haan (1970 [25]), $B\left(x\right)$ is a regularly varying function with index $1/\alpha$ at infinity, and by Karamata’s representation,

$$B\left(x\right)=x^{1/\alpha }c\left(x\right)exp\left\{{\int }_1^x{\displaystyle \frac{b_1\left(u\right)}{u}}\mathrm{d}u\right\}:=x^{1/\alpha }{l}_1\left(x\right),$$

(15)

where $l_1\left(x\right)$ is a slowly varying function, $\underset{x\to \infty }{lim}c\left(x\right)=c\in (0,\infty )$, and $\underset{x\to \infty }{lim}{b}_1\left(x\right)=0.$

Theorem 1.

Let $\left\{f_n\right\}$ be a sequence of positive nondecreasing numbers and $\{a_{n,k},1\le k\le n,n\ge 1\}$ be an array of weights satisfying (7). Assume that $\{X_n;n\ge 1\}$ is a sequence of e.i.i.d. random variables, $\mathbb{V}$ is continuous, and (13) holds with some $\alpha \in (0,2)$. For $\alpha \ge 1$, suppose that $\widehat{\mathbb{E}}$ is countably sub-additive and $\widehat{\mathbb{E}}{X}_1=\widehat{\epsilon }{X}_1$, and for $\alpha =1$, further suppose that $f_n{l}_1\left(n\right)\downarrow$. Set

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}\left\{\begin{array}{c}=0\mathrm{a}.\mathrm{s}.\mathbb{V},\\ =\infty \mathrm{a}.\mathrm{s}.\mathbf{\nu }\end{array}\right.$$

(16)

where $b=0$ for $0<\alpha <1$, and $b=\widehat{\mathbb{E}}{X}_1$ for $1\le \alpha <2$.

$${\displaystyle \sum _{n=1}^{\infty }}\mathbb{V}\left(|X_1|>B\left(n{f}_n\right)\right)\left\{\begin{array}{c}<\infty ,\\ =\infty .\end{array}\right.$$

(17)

$${\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n{f}_n}}\left\{\begin{array}{c}<\infty ,\\ =\infty .\end{array}\right.$$

(18)

Then, (17) is equivalent to (18), and (17) implies (16).

Theorem 2.

Under the conditions of Theorem 1,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\le {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbb{V},\end{array}$$

(19)

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\ge {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.\end{array}$$

(20)

This implies

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu },\end{array}$$

where b is defined by Theorem 1.

Remark 2.

Taking $a_{n,k}=1$ in Theorems 1 and 2, Theorems 1 and 2 become Theorems 4.1 and 4.2 of Wu and Jiang (2018 [17]). Therefore, Theorems 4.1 and 4.2 of Wu and Jiang (2018 [17]) are the special cases of Theorems 1 and 2, respectively, in this paper.

Remark 3.

Theorem 2 has results; we can obtain various forms LIL for a weighted sum by taking different forms of weights $a_{n,k}$. In particular, let h be a bounded variation function on $[0,1]$, with $h\left(1\right)>0$. Then, both $a_{n,k}:=h(k/n)$ and $a_{n,k}:=h\left(1\right)-h((k-1)/n)={\sum }_{i=k}^n(h(i/n)-h((i-1)/n))$ satisfy the condition (7). Therefore, we have 1 and 2.

Corollary 1.

Assume that $\{X_n;n\ge 1\}$ is a sequence of e.i.i.d. random variables, $\mathbb{V}$ is continuous, and (13) holds with some $\alpha \in (0,2)$. For $\alpha \ge 1$, suppose that $\widehat{\mathbb{E}}$ is countably sub-additive and $\widehat{\mathbb{E}}{X}_1=\widehat{\epsilon }{X}_1$. Let h be a bounded variation function on $[0,1]$ with $h\left(1\right)>0$. Then,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}h\left(\frac{k}{n}\right)(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\le {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbb{V},$$

and

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}h\left(\frac{k}{n}\right)(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\ge {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

This implies

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}h\left(\frac{k}{n}\right)(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu },\end{array}$$

where b is defined by Theorem 1.

Taking $a_{n,k}:=h\left(1\right)-h((k-1)/n)={\sum }_{i=k}^n(h(i/n)-h((i-1)/n))$ in Theorem 2, by ${\sum }_{k=1}^n{a}_{n,k}(X_k-b)={\sum }_{k=1}^n\left(h\left({\displaystyle \frac{k}{n}}\right)-h\left({\displaystyle \frac{k-1}{n}}\right)\right)(S_k-kb)$, we can immediately obtain the CLIL of weighted sums of partial sums as follows.

Corollary 2.

Under the conditions of Corollary 1,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}\left(h\left(\frac{k}{n}\right)-h\left(\frac{k-1}{n}\right)\right)(S_k-kb)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\le {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbb{V},$$

and

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}\left(h\left(\frac{k}{n}\right)-h\left(\frac{k-1}{n}\right)\right)(S_k-kb)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\ge {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

This implies

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}\left(h\left(\frac{k}{n}\right)-h\left(\frac{k-1}{n}\right)\right)(S_k-kb)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu },\end{array}$$

where b is defined by Theorem 1.

As another application of Theorem 2, we obtain the following CLIL for moving sums $Y_i:={\sum }_{k=1}^i{b}_{i-k+1}(X_k-b)$. Taking $a_{n,k}:={\sum }_{i=k}^n{b}_{i-k+1}$ in Theorem 2, by ${\sum }_{k=1}^n{a}_{n,k}(X_k-b)={\sum }_{i=1}^n{Y}_i$, we have the following corollary.

Corollary 3.

Assume that $\{X_n;n\ge 1\}$ is a sequence of e.i.i.d. random variables, $\mathbb{V}$ is continuous, and (13) holds with some $\alpha \in (0,2)$. For $\alpha \ge 1$, suppose that $\widehat{\mathbb{E}}$ is countably sub-additive and $\widehat{\mathbb{E}}{X}_1=\widehat{\epsilon }{X}_1$. Let $Y_i:={\sum }_{k=1}^i{b}_{i-k+1}(X_k-b)$, where $\{b_n;n\ge 1\}$ satisfies ${\sum }_{n=1}^{\infty }\left|b_n\right|<\infty$ and $b_1\ne 0$. Then,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{i=1}^n}{Y}_i\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\le {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbb{V},$$

and

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{i=1}^n}{Y}_i\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\ge {\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

This implies

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{i=1}^n}{Y}_i\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu },\end{array}$$

where b is defined by Theorem 1.

Proof of Theorem 1.

Without loss of generality, we can suppose that $f_n$ varies slowly. First, we prove that (17) ⇔ (18). Using the properties of regular variation (see Seneta (1976 [24]), we have

$$G\left(B\left(x\right)\right)\sim x^{-1},x\to \infty .$$

(21)

From (13) and (21),

$$\mathbb{V}\left(\right|X_1|>B\left(n{f}_n\right))=G\left(B\left(n{f}_n\right)\right)\sim {\displaystyle \frac{1}{n{f}_n}}.$$

Hence, (17) is equivalent to (18).

Now, we prove that (17)⇒(16).

Set $l_2\left(x\right)=cexp\left\{{\displaystyle {\int }_1^x\frac{b_1\left(u\right)}{u}\mathrm{d}u}\right\}$, where $c=\underset{x\to \infty }{lim}c\left(x\right),c\left(x\right)$ is defined by (15), and $b\left(x\right)=x^{1/\alpha }{l}_2\left(x\right)$. By (15), it is easy to see that

$$B\left(x\right)\sim b\left(x\right),x\to \infty .$$

Therefore, $B\left(n{f}_n\right)$ in (16)∼(17) can be replaced with $b\left(n{f}_n\right)$. Letting $a_n=b\left(n{f}_n\right)$, from the proof of Theorem 4.1 in Wu and Jiang (2018 [17]), the sequence $\left\{a_n\right\}$ satisfies the conditions of Lemma 2.

From the properties of slowly varying functions, if $l\left(x\right)$ is a slowly varying function, then for any $\beta >0$, there exists an increasing function $\phi \left(x\right)$ such that $l\left(x\right)x^{\beta }\sim \phi \left(x\right)$ and ${lim}_{x\to \infty }l\left(x\right)x^{\beta }=\infty$. Therefore, without loss of generality, we can suppose $l\left(x\right)x^{\beta }\uparrow \infty$ for any $\beta >0$. This implies

For $0<\alpha <1$, we have ${\displaystyle \frac{a_n}{n}\sim n^{1/\alpha -1}{f}_n^{1/\alpha }{l}_1\left(n\right)\uparrow \infty }$.

For $\alpha >1$, ${\displaystyle \frac{n}{a_n}}\sim {\displaystyle \frac{n^{1-1/\alpha }}{f_n^{1/\alpha }{l}_1\left(n\right)}}\uparrow$.

For $\alpha =1$, by assumption, ${\displaystyle \frac{n}{a_n}}\sim {\displaystyle \frac{1}{f_n{l}_1\left(n\right)}}\uparrow$.

Therefore, from Lemma 3 and the assumptions of Theorem 1, we obtain that (17) implies (16). □

Proof of Theorem 2.

For $\forall \epsilon >0$, let $f_n={\prod }_{j=1}^k{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n$. By (13) and (21), we obtain

$${\displaystyle \sum _{n=1}^{\infty }}\mathbb{V}\left(|X_1|>B\left(n{f}_n\right)\right)={\displaystyle \sum _{n=1}^{\infty }}G\left(B\left(n{f}_n\right)\right)\sim {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n}}<\infty .$$

Applying Theorem 1, we obtain

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}=0\mathrm{a}.\mathrm{s}.\mathbb{V}.$$

Thus, let ${\displaystyle \delta =\frac{\epsilon }{(1+\epsilon )\alpha }>0}$ in Lemma 1. For a sufficiently large n, by combining (15), we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}& =& {\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}{\displaystyle \frac{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n\right)}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\le {\displaystyle \frac{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n\right)}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\hfill \\ & =& {\displaystyle \frac{{\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n\right)}^{1/\alpha }{l}_1\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1+\epsilon }n\right)}{{\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{1/\alpha }{l}_1\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\hfill \\ & \le & {\left({lg}_{k+1}n\right)}^{(1+2\epsilon )/\alpha }\mathrm{a}.\mathrm{s}.\mathbb{V}.\hfill \end{array}$$

Therefore

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}\le {\mathrm{e}}^{{\displaystyle \frac{1+2\epsilon }{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbb{V}.$$

(22)

Similarly, for $\epsilon \in (0,1/2)$, let $f_n={\prod }_{j=1}^k{lg}_{j}n{lg}_{k+1}^{1-\epsilon }n.$ Then,

$${\displaystyle \sum _{n=1}^{\infty }}\mathbb{V}\left(|X_1|>B\left(n{f}_n\right)\right)\sim {\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1-\epsilon }n}}=\infty .$$

Applying Theorem 1, we have

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}=\infty \mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

Thus, there exist infinite indices n such that

$${\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}\ge 1\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

Let ${\displaystyle \delta =\frac{\epsilon }{(1-\epsilon )\alpha }>0}$ in Lemma 1. For these n, combining (15), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}& =& {\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{f}_n\right)}}{\displaystyle \frac{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1-\epsilon }n\right)}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\hfill \\ & \ge & {\displaystyle \frac{{\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1-\epsilon }n\right)}^{1/\alpha }{l}_1\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n{lg}_{k+1}^{1-\epsilon }n\right)}{{\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{1/\alpha }{l}_1\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\hfill \\ & \ge & {\left({lg}_{k+1}n\right)}^{{\displaystyle \frac{1-2\epsilon }{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.\hfill \end{array}$$

Therefore

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}}\right)}^{{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle {lg}_{k+2}n}}}}\ge {\mathrm{e}}^{{\displaystyle \frac{1-2\epsilon }{\alpha }}}\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

(23)

Since $\epsilon \in (0,1/2)$ is arbitrary, we obtain (19) and (20) from (22) and (23). □

5. Two Examples

In this section, we give two examples to show the limits in Theorem 2 with the various forms.

Example 2.

Assume that $X_1$ satisfies

$$G\left(x\right)=\mathbb{V}\left(|X_1|>x\right)={\displaystyle \frac{{\alpha }^{\beta }{(lgx)}^{\beta }}{x^{\alpha }}},\alpha \in (0,2),\beta \in \mathbb{R}.$$

It is easy to verify

$$B\left(x\right)=x^{{\displaystyle \frac{1}{\alpha }}}{(lgx)}^{{\displaystyle \frac{\beta }{\alpha }}}.$$

This implies that (21) holds and

$$B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)\sim n^{{\displaystyle \frac{1}{\alpha }}}{(lgn)}^{{\displaystyle \frac{\beta }{\alpha }}}{\left({\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{{\displaystyle \frac{1}{\alpha }}}.$$

Hence, by Theorem 2,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{n^{\frac{1}{\alpha }}{(lgn)}^{\frac{\beta }{\alpha }}{\left({\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{\frac{1}{\alpha }}}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}},\mathrm{a}.\mathrm{s}.\mathbf{\nu },$$

where $a_{n,k}$ and b are defined by Theorem 2.

In particular, let $k=0$; then,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{n^{\frac{1}{\alpha }}{(lgn)}^{\frac{\beta }{\alpha }}}}\right)}^{{\displaystyle \frac{1}{lglgn}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}},\mathrm{a}.\mathrm{s}.\mathbf{\nu }$$

or

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{n^{\frac{1}{\alpha }}}}\right)}^{{\displaystyle \frac{1}{lglgn}}}={\mathrm{e}}^{{\displaystyle \frac{1+\beta }{\alpha }}},\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

Example 3.

Assume that $X_1$ satisfies

$$G\left(x\right)=\mathbb{V}\left(\right|X_1|>x)\sim {\displaystyle \frac{exp\left(\beta {(lgx)}^r\right)}{x^{\alpha }}},r\in (0,1),\beta \in \mathbb{R},\alpha \in (0,2).$$

It is easy to check

$$B\left(x\right)=x^{{\displaystyle \frac{1}{\alpha }}}exp\left({\displaystyle \frac{\beta {(lgx)}^r}{{\alpha }^{r+1}}}\right).$$

This implies that (21) holds and

$$B\left(n{\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)\sim n^{{\displaystyle \frac{1}{\alpha }}}{\left({\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{{\displaystyle \frac{1}{\alpha }}}exp\left({\displaystyle \frac{\beta {(lgn)}^r}{{\alpha }^{r+1}}}\right).$$

Hence, by Theorem 2,

$$\underset{n\to \infty }{lim\; sup}{\left({\displaystyle \frac{\left|{\displaystyle \sum _{k=1}^n}{a}_{n,k}(X_k-b)\right|}{n^{\frac{1}{\alpha }}{\left({\displaystyle \prod _{j=1}^k}{lg}_{j}n\right)}^{\frac{1}{\alpha }}exp\left(\frac{\beta {(lgn)}^r}{{\alpha }^{r+1}}\right)}}\right)}^{{\displaystyle \frac{1}{{lg}_{k+2}n}}}={\mathrm{e}}^{{\displaystyle \frac{1}{\alpha }}},\mathrm{a}.\mathrm{s}.\mathbf{\nu }.$$

6. Conclusions

This study establishes Chover’s law of the iterated logarithm for weighted sums of independent and identically distributed random variables under sub-linear expectation spaces. The results enrich the limit theory in non-additive probability frameworks and provide a rigorous theoretical foundation for statistical inference involving distribution uncertainty.

The findings offer broad applicability, particularly in fields such as financial risk measurement, econometric modeling, and complex systems analysis, where uncertainty and imprecise probabilistic structures are inherent. For instance, the proposed limit theorems can help characterize extreme behaviors or tail properties in uncertain financial returns or economic indicators, supporting better risk predictions and decision-making under ambiguity.

Moreover, this work is conceptually aligned with recent advances in generalized fuzzy set theories, especially in handling uncertainty and imprecision.

For example, similar to how fuzzy upper Mandelbrot sets capture fractal structures under fuzzy perturbations, our model deals with probabilistic uncertainty in limit theorems, suggesting potential cross-disciplinary applications in uncertain dynamical systems.

The structure of circular Pythagorean fuzzy sets (CPFSs) and circular q-rung orthopair fuzzy sets (Cq-ROFSs), which deal with directional and periodic uncertainty in membership and non-membership grades, resonates with our approach to handling distributional uncertainty in random variables. A promising future research direction would be to incorporate sub-linear expectation into fuzzy set-based decision systems, especially those involving circular or periodic information, such as in environmental modeling or signal processing under uncertainty.

We anticipate that the methodology and results presented herein can be further integrated with these fuzzy frameworks to develop more robust hybrid models for uncertainty reasoning, enhancing both theoretical and applied research in stochastic and fuzzy systems.