Bootstrapping Average Value at Risk of Single and Collective Risks

Abstract

Almost sure bootstrap consistency of the blockwise bootstrap for the Average Value at Risk of single risks is established for strictly stationary $\beta$-mixing observations. Moreover, almost sure bootstrap consistency of a multiplier bootstrap for the Average Value at Risk of collective risks is established for independent observations. The main results rely on a new functional delta-method for the almost sure bootstrap of uniformly quasi-Hadamard differentiable statistical functionals, to be presented here. The latter seems to be interesting in its own right.

1. Introduction

One of the most popular risk measures in practice is the so-called Average Value at Risk which is also referred to as Expected Shortfall (see Acerbi and Szekely (2014); Acerbi and Tasche (2002a, 2002b); Emmer et al. (2015) and references therein). For a fixed level $\alpha \in (0,1)$, the corresponding Average Value at Risk is the map ${\mathrm{AV}@\mathrm{R}}_{\alpha }:L^1\to \mathbb{R}$ defined by ${\mathrm{AV}@\mathrm{R}}_{\alpha }(X):={\mathcal{R}}_{\alpha }(F_X)$, where $F_X$ refers to the distribution function of X, $L^1$ is the usual $L^1$-space associated with some atomless probability space, and

$${\mathcal{R}}_{\alpha }(F):={\int }_0^1{F}^{\leftarrow }(s)d{g}_{\alpha }(s)=-{\int }_{-\infty }^0{g}_{\alpha }(F(x))dx+{\int }_0^{\infty }\left(1-g_{\alpha }(F(x))\right)dx$$

(1)

for any $F\in {\mathbf{F}}_1$ with ${\mathbf{F}}_1$ the set of the distribution functions $F_X$ of all $X\in L^1$. Here, $g_{\alpha }(t):=\frac{1}{1-\alpha }max\{t-\alpha ;0\}$ and $F^{\leftarrow }(s):=inf\{x\in \mathbb{R}:F(x)\ge s\}$ denotes the left-continuous inverse of F. The statistical functional ${\mathcal{R}}_{\alpha }:{\mathbf{F}}_1\to \mathbb{R}$ is sometimes referred to as risk functional associated with ${\mathrm{AV}@\mathrm{R}}_{\alpha }$. Note that ${\mathrm{AV}@\mathrm{R}}_{\alpha }(X)=\mathbb{E}[X|X\ge F_X^{\leftarrow }(\alpha )]$ when $F_X$ is continuous at $F_X^{\leftarrow }(\alpha )$.

In this article, we mainly focus on bootstrap methods for the Average Value at Risk. Before doing so, we briefly review nonparametric estimation techniques and asymptotic results for the Average Value at Risk. Given identically distributed observations $X_1,\dots ,X_n$ ($,X_{n+1},\dots$) on some probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with unknown marginal distribution $F\in {\mathbf{F}}_1$, a natural estimator for ${\mathcal{R}}_{\alpha }(F)$ is the empirical plug-in estimator

$${\mathcal{R}}_{\alpha }({\widehat{F}}_n)={\int }_0^1{\widehat{F}}_n^{\leftarrow }(s)d{g}_{\alpha }(s)={\displaystyle \sum _{i=1}^n}\left\{g_{\alpha }\left(\frac{i}{n}\right)-g_{\alpha }\left(\frac{i-1}{n}\right)\right\}{X}_{i:n},$$

(2)

where ${\widehat{F}}_n:=\frac{1}{n}{\sum }_{i=1}^n{𝟙}_{[X_i,\infty )}$ is the empirical distribution function of $X_1,\dots ,X_n$ and $X_{1:n},\dots ,X_{n:n}$ refer to the order statistics of $X_1,\dots ,X_n$. The second representation in Equation (2) shows that ${\mathcal{R}}_{\alpha }({\widehat{F}}_n)$ is a specific L-statistic which was already mentioned in Acerbi (2002); Acerbi and Tasche (2002a); Jones and Zitikis (2003).

In particular, if the underlying sequence ${(X_i)}_{i\in \mathbb{N}}$ is strictly stationary and ergodic, classical results of van Zwet (1980) and Gilat and Helmers (1997) show that ${\mathcal{R}}_{\alpha }({\widehat{F}}_n)$ converges $\mathbb{P}$-almost surely to ${\mathcal{R}}_{\alpha }(F)$ as $n\to \infty$, i.e., that strong consistency holds. If $X_1,X_2,\dots$ are i.i.d. and F has a finite second moment and takes the value $\alpha$ only once, then a result of Stigler ((Stigler 1974, Theorems 1–2)) yields the asymptotic distribution of the estimation error:

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n)-{\mathcal{R}}_{\alpha }(F)\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_F^2},$$

(3)

where ${\sigma }_F^2:=\int \int g_{\alpha }^{\prime }(F(x_0))\Gamma (x_0,x_1)g_{\alpha }^{\prime }(F(x_1))d{x}_{0}d{x}_1$ with $\Gamma (x_0,x_1):=F(x_0\wedge x_1)(1-F(x_0\vee x_1))+{\sum }_{i=0}^1{\sum }_{k=2}^{\infty }\mathbb{C}\mathrm{ov}({𝟙}_{\{X_1\le x_i\}},{𝟙}_{\{X_k\le x_{1-i}\}})$, $g_{\alpha }^{\prime }:=\frac{1}{1-\alpha }{𝟙}_{(\alpha ,1]}$, and ⇝ refers to convergence in distribution (see also Shorack 1972; Shorack and Wellner 1986). In fact, for independent $X_1,X_2,\dots$ the second summand in the definition of $\Gamma (x_0,x_1)$ vanishes. Results of Beutner and Zähle (2010) show that Equation (3) still holds if ${(X_i)}_{i\in \mathbb{N}}$ is strictly stationary and $\alpha$-mixing with mixing coefficients $\alpha (i)=\mathcal{O}(i^{-\theta })$ and ${lim}_{x\to \infty }(1-F(x))x^{2\theta /(\theta -1)}<\infty$ for some $\theta >1+\sqrt{2}$. Tsukahara (2013) obtained the same result. A similar result can also be derived from an earlier work by Mehra and Rao (1975), but under a faster decay of the mixing coefficients and under an additional assumption on the dependence structure. We emphasize that the method of proof proposed by Beutner and Zähle is rather flexible, because it easily extends to other weak and strong dependence concepts and other risk measures (see Beutner et al. 2012; Beutner and Zähle 2010, 2016; Krätschmer et al. 2013; Krätschmer and Zähle 2017).

Even in the i.i.d. case the asymptotic variance ${\sigma }_F^2$ depends on F in a fairly complex way. For the approximation of the distribution of $\sqrt{n}({\mathcal{R}}_{\alpha }({\widehat{F}}_n)-{\mathcal{R}}_{\alpha }(F))$, bootstrap methods should thus be superior to the method of estimating ${\sigma }_F^2$. However, to the best of our knowledge, theoretical investigations of the bootstrap for the Average Value at Risk seem to be rare. According to Gribkova (2016), a result of Gribkova (2002) yields bootstrap consistency for Efron’s bootstrap when $X_1,X_2,\dots$ are i.i.d, while Theorem 3 of Helmers et al. (1990) seems not to cover the Average Value at Risk, because there the function J (which plays the role of $g_{\alpha }^{\prime }$) is assumed to be Lipschitz continuous. In these articles, bootstrap consistency is typically proved by first proving consistency of the bootstrap variance and then using this result by showing that upper bounds for the difference between the sampling distribution and the bootstrap distribution converge to zero. Employing different techniques, Beutner and Zähle (2016) established bootstrap consistency in probability for the multiplier bootstrap when $X_1,X_2,\dots$ are i.i.d. as well as bootstrap consistency in probability for the circular bootstrap when $X_1,X_2,\dots$ are strictly stationary and $\beta$-mixing with mixing coefficients $\beta (i)=\mathcal{O}(i^{-b})$ and ${\int |x|}^{p}dF(x)<\infty$ for some $p>2$ and $b>p/(p-2)$. Recently, Sun and Cheng (2018) established bootstrap consistency in probability for the moving blocks bootstrap when $X_1,X_2,\dots$ are strictly stationary and $\alpha$-mixing with mixing coefficients $\alpha (i)\le c{\delta }^i$ and ${\int |x|}^{p}dF(x)<\infty$ for some $p>4$, $c>0$ and $\delta \in (0,1)$. Strictly speaking, Sun and Cheng did not consider the Average Value at Risk (Expected Shortfall) but the Tail Conditional Expectation in the sense of Acerbi and Tasche (2002a, 2002b).

The contribution of the article at hand is twofold. First, we extend the results of Beutner and Zähle (2016) on the Average Value at Risk from bootstrap consistency in probability to bootstrap consistency almost surely. Second, we establish bootstrap consistency for the Average Value at Risk of collective risks, i.e., for ${\mathcal{R}}_{\alpha }(F^{\ast m})$ and more general expressions.

The rest of the article is organized as follows. In Section 2, we present and illustrate our main results which are proved in Section 3. Section 3 is followed by the conclusions. The proofs of Section 3 rely on a new functional delta-method for the almost sure bootstrap which seems to be interesting in its own right and which is presented in Appendix B. Roughly speaking, the (functional) delta method studies properties of particular estimators for quantities of the form $H(\theta )$. Here, H is a known functional, such as the Average Value at Risk functional, and $\theta$ is a possibly infinite dimensional parameter, such as an unknown distribution function. The particular estimators covered by the (functional) delta method are of the form $H({\widehat{T}}_n)$ where ${\widehat{T}}_n$ is an estimator for $\theta$. In general and in the particular application considered here, the appeal of the (functional) delta method lies in the fact that, once “differentiability” of H (here, the Average Value at Risk functional) is established, the asymptotic error distribution of $H({\widehat{T}}_n)$ can immediately be derived from the asymptotic error distribution of ${\widehat{T}}_n$ (here ${\widehat{F}}_n$). This also applies to the (functional) delta method for the bootstrap where bootstrap consistency of the bootstrapped version of $H({\widehat{T}}_n)$ will follow from the respective property of the bootstrapped version of ${\widehat{T}}_n$ (here ${\widehat{F}}_n$). Thus, if in financial or actuarial applications the data show dependencies for which the asymptotic error distribution and/or bootstrap consistency of plug-in estimators for the Average Value at Risk have not been established yet, it would be enough to check if for these dependencies the asymptotic error distribution and/or bootstrap consistency of ${\widehat{F}}_n$ is known; thanks to the (functional) delta method the Average Value at Risk functional would inherit these properties. In Appendix A.1, we give results on convergence in distribution for the open-ball $\sigma$-algebra which are needed for the main results, and in Appendix A.2 we prove a delta-method for uniformly quasi-Hadamard differentiable maps that is the basis for the method of Appendix B. Readers interested in these methods used to prove the main results might wish to first work through Appendix A and Appendix B before reading Section 2 and Section 3.

2. Main Results

2.1. The Case of i.i.d. Observations

Keep the notation of Section 1. Assume that ${(X_i)}_{i\in \mathbb{N}}$ is a sequence of i.i.d. real-valued random variables on some probability space $(\Omega ,\mathcal{F},\mathbb{P})$ with distribution function F. Let ${\widehat{F}}_n:=\frac{1}{n}{\sum }_{i=1}^n{𝟙}_{[X_i,\infty )}$ and $(W_{ni})$ be a triangular array of nonnegative real-valued random variables on another probability space $({\Omega }^{\prime },{\mathcal{F}}^{\prime },{\mathbb{P}}^{\prime })$ such that one of the following two settings is met.

S1.

The random vector $(W_{n1},\dots ,W_{nn})$ is multinomially distributed according to the parameters n and $p_1=\cdots =p_n:=1/n$ for every $n\in \mathbb{N}$.

S2.

$W_{ni}=Y_i/{\overline{Y}}_n$ for every $i=1,\dots ,n$ and $n\in \mathbb{N}$, where ${\overline{Y}}_n:=\frac{1}{n}{\sum }_{j=1}^n{Y}_j$ and $(Y_j)$ is any sequence of nonnegative i.i.d. random variables on $({\Omega }^{\prime },{\mathcal{F}}^{\prime },{\mathbb{P}}^{\prime })$ with ${\int }_0^{\infty }{\mathbb{P}}^{\prime }{[Y_1>t]}^{1/2}dt<\infty$ and $\mathbb{V}{\mathrm{ar}}^{\prime }{[Y_1]}^{1/2}={\mathbb{E}}^{\prime }[Y_1]>0$.

Let $(\overline{\Omega },\overline{\mathcal{F}},\overline{\mathbb{P}}):=(\Omega \times {\Omega }^{\prime },\mathcal{F}\otimes {\mathcal{F}}^{\prime },\mathbb{P}\otimes {\mathbb{P}}^{\prime })$ and ${\widehat{F}}_n^{\ast }(\omega ,{\omega }^{\prime }):=\frac{1}{n}{\sum }_{i=1}^n{W}_{ni}({\omega }^{\prime }){𝟙}_{[X_i(\omega ),\infty )}$. Setting S1. is nothing but Efron’s boostrap (Efron 1979). If in Setting S2. the distribution of $Y_1$ is the exponential distribution with parameter 1, then the resulting scheme is in line with the Bayesian bootstrap of Rubin (1981). Let ${\sigma }_F^2:=\int \int g_{\alpha }^{\prime }(F(x_0))\Gamma (x_0,x_1)g_{\alpha }^{\prime }(F(x_1))d{x}_{0}d{x}_1$ with $\Gamma (x_0,x_1):=F(x_0\wedge x_1)(1-F(x_0\vee x_1))$.

Theorem 1.

In the setting above assume that $\int {\varphi }^{2}dF<\infty$ for some continuous function $\varphi :\mathbb{R}\to [1,\infty )$ with $\int 1/\varphi (x)dx<\infty$ (in particular $F\in {\mathbf{F}}_1$), and that F takes the value α only once. Then

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n)-{\mathcal{R}}_{\alpha }(F)\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_F^2}$$

(4)

and

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n^{\ast }(\omega ,·))-{\mathcal{R}}_{\alpha }({\widehat{F}}_n(\omega ))\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_F^2},\mathbb{P}-a.e.\omega .$$

(5)

Theorem 1 is a special case of Corollary 1 below. For the bootstrap Scheme S1. the result of Theorem 1 can be also deduced from Theorem 7 in Gribkova (2002). According to Gribkova (2016), Condition (1) of this theorem is satisfied if there are $0=a_0<a_1<\cdots <a_k=1$ for some $k\in \mathbb{N}$ such that J is Hölder continuous on each interval $(a_{i-1},a_i)$, $1\le i\le k$, and the measure $d{F}^{-1}$ has no mass at the points $a_1,\dots ,a_{k-1}$. For the bootstrap Scheme S2. the result seems to be new.

We now consider the collective risk model. Let ${(X_i)}_{i\in \mathbb{N}}$ and ${\widehat{F}}_n$ be as above, and let $p={(p_k)}_{k\in {\mathbb{N}}_0}$ be the counting density of a distribution on ${\mathbb{N}}_0$. Let $\mathbf{F}$ denote the set of all distribution functions on $\mathbb{R}$, and consider the functional ${\mathcal{C}}_p:\mathbf{F}\to \mathbf{F}$ defined by ${\mathcal{C}}_p(F):={\sum }_{k=0}^{\infty }{p}_k{F}^{\ast k}$, where $F^{\ast k}$ refers to the k-fold convolution of F, i.e., $F^{\ast 0}:={𝟙}_{[0,\infty )}$ and $F^{\ast k}(x):=\int F(x-x_{k-1})d{F}^{\ast (k-1)}(x_{k-1})=\int \cdots \int F(x-x_{k-1}-\cdots -x_1)dF(x_1)\cdots dF(x_{k-1})$ for $k\in \mathbb{N}$. If $p_m=1$ for some $m\in {\mathbb{N}}_0$, then ${\mathcal{C}}_p(F)=F^{\ast m}$. Let ${\sigma }_{p,F}^2:=\int \int \int \int g_{\alpha }^{\prime }(F(x_0))\Gamma (x_0-y_0,x_1-y_1)g_{\alpha }^{\prime }(F(x_1))d{H}_{p,F}(y_0)d{H}_{p,F}(y_1)d{x}_{0}d{x}_1$ with $\Gamma (z_0,z_1):=F(z_0\wedge z_1)(1-F(z_0\vee z_1))$ and $H_{p,F}:={\sum }_{k=1}^{\infty }k{p}_k{F}^{\ast (k-1)}$.

Theorem 2.

In the setting above assume that ${\int |x|}^{2\lambda }dF(x)<\infty$ for some $\lambda >1$ (in particular $F\in {\mathbf{F}}_1$) and ${\sum }_{k=1}^{\infty }{p}_k{k}^{1+\lambda }<\infty$, and that ${\mathcal{C}}_p(F)$ takes the value α only once. Then,

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\mathcal{C}}_p({\widehat{F}}_n))-{\mathcal{R}}_{\alpha }({\mathcal{C}}_p(F))\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_{p,F}^2}$$

(6)

and

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\mathcal{C}}_p({\widehat{F}}_n^{\ast }(\omega ,·)))-{\mathcal{R}}_{\alpha }({\mathcal{C}}_p({\widehat{F}}_n(\omega )))\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_{p,F}^2},\mathbb{P}-a.e.\omega .$$

(7)

Theorem 2 is a special case of Corollary 4 below. Lauer and Zähle (2015, 2017) derive the asymptotic distribution as well as almost sure bootstrap consistency for the Average Value at Risk (and more general risk measures) of $F^{\ast m_n}$ when $m_n/n$ is asymptotically constant, but we do not know any result in the existing literature which is comparable to that of Theorem 2.

2.2. The Case of $\beta$-Mixing Observations

Keep the notation of Section 1. Assume that ${(X_i)}_{i\in \mathbb{N}}$ is a strictly stationary sequence of $\beta$-mixing random variables on $(\Omega ,\mathcal{F},\mathbb{P})$ with distribution function F. As before let ${\widehat{F}}_n:=\frac{1}{n}{\sum }_{i=1}^n{𝟙}_{[X_i,\infty )}$. Let $({\ell }_n)$ be a sequence of integers such that ${\ell }_n\nearrow \infty$ as $n\to \infty$, and ${\ell }_n<n$ for all $n\in \mathbb{N}$. Set $k_n:=\lceil n/{\ell }_n\rceil$ for all $n\in \mathbb{N}$. Let ${(I_{nj})}_{n\in \mathbb{N},1\le j\le k_n}$ be a triangular array of random variables on $({\Omega }^{\prime },{\mathcal{F}}^{\prime },{\mathbb{P}}^{\prime })$ such that $I_{n1},\dots ,I_{n{k}_n}$ are i.i.d. according to the uniform distribution on $\{1,\dots ,n-{\ell }_n+1\}$ for every $n\in \mathbb{N}$. Let $(\overline{\Omega },\overline{\mathcal{F}},\overline{\mathbb{P}}):=(\Omega \times {\Omega }^{\prime },\mathcal{F}\otimes {\mathcal{F}}^{\prime },\mathbb{P}\otimes {\mathbb{P}}^{\prime })$ and ${\widehat{F}}_n^{\ast }(\omega ,{\omega }^{\prime }):=\frac{1}{n}{\sum }_{i=1}^n{W}_{ni}({\omega }^{\prime }){𝟙}_{[X_i(\omega ),\infty )}$ with

$$W_{ni}({\omega }^{\prime }):={\displaystyle \sum _{j=1}^{k_n-1}}{𝟙}_{\{I_{nj}\le i\le I_{nj}+{\ell }_n-1\}}({\omega }^{\prime })+{𝟙}_{\{I_{n{k}_n}\le i\le I_{n{k}_n}+(n-(k_n-1){\ell }_n)-1\}}({\omega }^{\prime }).$$

(8)

Note that the sequence $(X_i)$ and the triangular array $(W_{ni})$ regarded as families of random variables on the product space $(\overline{\Omega },\overline{\mathcal{F}},\overline{\mathbb{P}}):=(\Omega \times {\Omega }^{\prime },\mathcal{F}\otimes {\mathcal{F}}^{\prime },\mathbb{P}\otimes {\mathbb{P}}^{\prime })$ are independent. At an informal level, this means that, given a sample $X_1,\dots ,X_n$, we pick $k_n-1$ blocks of length ${\ell }_n$ and one block of length $n-(k_n-1){\ell }_n$ in the sample $X_1,\dots ,X_n$, where the start indices $I_{n1},I_{n2},\dots ,I_{n{k}_n}$ are chosen independently and uniformly in the set of indices $\{1,\dots ,n-{\ell }_n+1\}$:

block 1:	$X_{I_{n1}},X_{I_{n1}+1},\dots ,X_{I_{n1}+{\ell }_n-1}$
block 2:	$X_{I_{n2}},X_{I_{n2}+1},\dots ,X_{I_{n2}+{\ell }_n-1}$
	⋮
block $k_n-1$:	$X_{I_{n(k_n-1)}},X_{I_{n(k_n-1)}+1},\dots ,X_{I_{n(k_n-1)}+{\ell }_n-1}$
block $k_n$:	$X_{I_{n{k}_n}},X_{I_{n{k}_n}+1},\dots ,X_{I_{n{k}_n}+(n-(k_n-1){\ell }_n)-1}$.

The bootstrapped empirical distribution function ${\widehat{F}}_n^{\ast }$ is then defined to be the distribution function of the discrete probability measure with atoms $X_1,\dots ,X_n$ carrying masses $W_{n1},\dots ,W_{nn}$, respectively, where $W_{ni}$ specifies the number of blocks which contain $X_i$. This is known as the blockwise bootstrap (see, e.g., Bühlmann (1994, 1995) and references therein). Assume that the following assertions hold:

A1.

$\int {\varphi }^{p}dF<\infty$ for some $p>4$ (in particular $F\in {\mathbf{F}}_1$).

A2.

The sequence of random variables $(X_i)$ is strictly stationary and $\beta$-mixing with mixing coefficients $({\beta }_i)$ satisfying ${\beta }_i\le c{\delta }^i$ for some constants $c>0$ and $\delta \in (0,1)$.

A3.

The block length ${\ell }_n$ satisfies ${\ell }_n=\mathcal{O}(n^{\gamma })$ for some $\gamma \in (0,1/2)$.

Let ${\widehat{C}}_n:={\mathbb{E}}^{\prime }[{\widehat{F}}_n^{\ast }]=\frac{1}{n}{\sum }_{i=1}^n{w}_{ni}{𝟙}_{[X_i,\infty )}$ with $w_{ni}:={\mathbb{E}}^{\prime }[W_{ni}]$, and note that

$$w_{ni}=\left\{\begin{array}{ccc}{k}_n\frac{i}{n-{\ell }_n+1}\hfill & ,\hfill & i=1,\dots ,n-(k_n-1){\ell }_n\hfill \\ (k_n-1)\frac{i}{n-{\ell }_n+1}+\frac{n-(k_n-1){\ell }_n}{n-{\ell }_n+1}\hfill & ,\hfill & i=n-(k_n-1){\ell }_n+1,\dots ,{\ell }_n\hfill \\ (k_n-1)\frac{{\ell }_n}{n-{\ell }_n+1}+\frac{n-(k_n-1){\ell }_n}{n-{\ell }_n+1}=\frac{n}{n-{\ell }_n+1}\hfill & ,\hfill & i={\ell }_n+1,\dots ,n-{\ell }_n\hfill \\ (k_n-1)\frac{n-i+1}{n-{\ell }_n+1}+\frac{2n-k_n{\ell }_n-i+1}{n-{\ell }_n+1}\hfill & ,\hfill & i=n-{\ell }_n+1,\dots ,n-(k_n{\ell }_n-n)\hfill \\ (k_n-1)\frac{n-i+1}{n-{\ell }_n+1}\hfill & ,\hfill & i=n-(k_n{\ell }_n-n)+1,\dots ,n\hfill \end{array}\right.$$

(9)

which can be verified easily. Let ${\sigma }_F^2:=\int \int g_{\alpha }^{\prime }(F(x_0))\Gamma (x_0,x_1)g_{\alpha }^{\prime }(F(x_1))d{x}_{0}d{x}_1$ with $\Gamma (x_0,x_1):=F(x_0\wedge x_1)(1-F(x_0\vee x_1))+{\sum }_{i=0}^1{\sum }_{k=2}^{\infty }\mathbb{C}\mathrm{ov}({𝟙}_{\{X_1\le x_i\}},{𝟙}_{\{X_k\le x_{1-i}\}})$.

Theorem 3.

In the setting above (in particular under A1.–A3.) assume that F takes the value α only once. Then, we have

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n)-{\mathcal{R}}_{\alpha }(F)\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_F^2}$$

(10)

and

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n^{\ast }(\omega ,·))-{\mathcal{R}}_{\alpha }({\widehat{C}}_n(\omega ))\right)⇝Z\sim {\mathrm{N}}_{0,{\sigma }_F^2},\mathbb{P}-a.e.\omega .$$

(11)

Theorem 3 is a special case of Corollary 1 below. To the best of our knowledge, there does not yet exist any result on almost sure bootstrap consistency for the Average Value at Risk when the underlying data are dependent.

2.3. Applications

2.3.1. Bootstrapping the Down Side Risk of an Asset Price

Let ${(A_i)}_{i\in {\mathbb{N}}_0}$ be the price process of an asset. Let us assume that it is induced by an initial state $A_0\in {\mathbb{R}}_{+}$ and a sequence of ${\mathbb{R}}_{+}$-valued i.i.d. random variables ${(R_i)}_{i\in \mathbb{N}}$ via $A_i:=R_i{A}_{i-1}$, $i\in \mathbb{N}$. Here, $R_i$ is the return of the asset in between time $i-1$ and time i. For instance, if $A_0,A_1,A_2,\dots$ are the observations of a time-continuous Black–Scholes–Merton model with drift $\mu$ and volatility $\sigma$ at the points of the time grid $\{0,h,2h,\dots \}$, then the distribution of $R_1$ is the log-normal distribution with parameters $(\mu -{\sigma }^2/2)h$ and ${\sigma }^{2}h$. However, the adequacy of a specific parametric model is usually hard to verify. For this reason, we do not restrict ourselves to any particular parametric structure for the dynamics of ${(R_i)}_{i\in \mathbb{N}}$.

Let us assume that we can observe the asset prices $A_0,\dots ,A_n$ up to time n, and that we are interested in the Average Value at Risk at level $\alpha$ of the negative price change $A_n-A_{n+1}$ (which specifies the down side risk of the asset) in between time n and $n+1$. That is, since for any $a_0,\dots ,a_n\in {\mathbb{R}}_{+}$ the unconditional distribution of $(1-R_{n+1})a_n$ coincides with the factorized conditional distribution of $A_n-A_{n+1}=(1-R_{n+1})A_n$ given $(A_0,\dots ,A_n)=(a_0,\dots ,a_n)$, we are in fact interested in ${\mathcal{R}}_{\alpha }(F)={\mathrm{AV}@\mathrm{R}}_{\alpha }(X)$ for the distribution function F of $X:=(1-R_{n+1})a$ for any fixed $a\in {\mathbb{R}}_{+}$. As the random variables $X_1:=(1-R_1)a,\dots ,X_n:=(1-R_n)a$ are i.i.d. copies of X, we can use ${\mathcal{R}}_{\alpha }({\widehat{F}}_n)$ as an estimator for ${\mathcal{R}}_{\alpha }(F)$ and derive from Equation (4) an asymptotic confidence interval at a given level $\tau \in (0,1)$ for ${\mathcal{R}}_{\alpha }(F)$ where one has to estimate ${\sigma }_F^2$ by $\int \int g_{\alpha }^{\prime }({\widehat{F}}_n(x_0)){\widehat{F}}_n(x_0\wedge x_1)(1-{\widehat{F}}_n(x_0\vee x_1))g_{\alpha }^{\prime }({\widehat{F}}_n(x_1))d{x}_{0}d{x}_1$. As the estimator for ${\sigma }_F^2$ depends on ${\widehat{F}}_n$ in a somewhat complex way, the bootstrap confidence interval

$$\left[{\mathcal{R}}_{\alpha }\left({\widehat{F}}_n(\omega )\right)-\frac{1}{\sqrt{n}}{\widehat{q}}_{1-\tau /2}^{\ast }(\omega ),{\mathcal{R}}_{\alpha }\left({\widehat{F}}_n(\omega )\right)-\frac{1}{\sqrt{n}}{\widehat{q}}_{\tau /2}^{\ast }(\omega )\right]$$

(12)

at level $\tau$ derived from Equations (4) and (5) is supposed to have a slightly better performance. Here, ${\widehat{q}}_t^{\ast }(\omega )$ denotes a t-quantile of (a Monte Carlo approximation of) the distribution of the left-hand side in Equation (5) for fixed $\omega$. For Equations (4) and (5) it suffices to assume that $\mathbb{E}[|R_1{|}^{2+\epsilon }]<\infty$ for some arbitrarily small $\epsilon >0$.

2.3.2. Bootstrapping the Total Risk Premium in Insurance Models

In actuarial mathematics, the collective risk model is frequently used for modeling the total claim distribution of an insurance collective. If the counting density $p={(p_k)}_{k\in {\mathbb{N}}_0}$ corresponds to the distribution of the random number N of claims caused by the whole collective within one insurance period, and if $X_1,\dots ,X_N$ ($,X_{N+1},\dots$) denote the i.i.d. sizes of the corresponding claims with marginal distribution F, then ${\mathcal{C}}_p(F)$ is the distribution of the total claim ${\sum }_{i=1}^N{X}_i$ (the latter sum is set to 0 if $N=0$). Now, ${\mathcal{R}}_{\alpha }({\mathcal{C}}_p(F))$ is a suitable insurance premium for the whole collective when the Average Value at Risk at level $\alpha$ is considered to be a suitable premium principle.

Assume that p is known, for instance $p_m=1$ for some fixed $m\in \mathbb{N}$, and let $X_1,\dots ,X_n$ be observed historical (i.i.d.) claims with n large. On the one hand, the construction of an exact confidence interval for ${\mathcal{R}}_{\alpha }({\mathcal{C}}_p(F))$ at level $\tau \in (0,1)$ based on $X_1,\dots ,X_n$ is hardly possible. Likewise, the performance of an asymptotic confidence interval at level $\tau$ derived from Equation (6) with (nonparametrically) estimated ${\sigma }_{p,F}^2$ is typically only moderate. Take into account that ${\sigma }_{p,F}^2$ depends on the unknown F in a fairly complex way. On the other hand, the bootstrap confidence interval

$$\left[{\mathcal{R}}_{\alpha }\left({\mathcal{C}}_p({\widehat{F}}_n(\omega ))\right)-\frac{1}{\sqrt{n}}{\widehat{q}}_{1-\tau /2}^{\ast }(\omega ),{\mathcal{R}}_{\alpha }\left({\mathcal{C}}_p({\widehat{F}}_n(\omega ))\right)-\frac{1}{\sqrt{n}}{\widehat{q}}_{\tau /2}^{\ast }(\omega )\right]$$

at level $\tau$ derived from Equation (7) should have a better performance. Here, ${\widehat{q}}_t^{\ast }(\omega )$ denotes a t-quantile of (a Monte Carlo approximation of) the distribution of the left-hand side in Equation (7) for fixed $\omega$.

Note that Theorem 2 ensures that Equations (6) and (7) hold true when the marginal distribution F of the $X_i$ is any log-normal distribution, any Gamma distribution, any Pareto distribution with tail index greater than 2, or any convex combination of one of these distributions with the Dirac measure ${\delta }_0$, and the counting density p corresponds to any Dirac measure with atom in $\mathbb{N}$, any binomial distribution, any Poisson distribution, or any geometric distribution. The former distributions are classical examples for the single claim distribution and the latter distributions are classical examples for the claim number distribution.

3. Proofs of Main Results

Here, we prove the results of Section 2. In fact, Theorems 1–3 are special cases of Corollaries 1 and 4. The latter corollaries are proved with the help of the technique introduced in Appendix B.2, which in turn avails the concept of uniform quasi-Hadamard differentiability (see Definition A1 in Appendix B.1).

Keep the notation introduced in Section 1. Let $\mathbf{D}$ be the space of all cádlág functions v on $\mathbb{R}$ with finite sup-norm ${\parallel v\parallel }_{\infty }:={sup}_{t\in \mathbb{R}}|v(t)|$, and $\mathcal{D}$ be the $\sigma$-algebra on $\mathbf{D}$ generated by the one-dimensional coordinate projections ${\pi }_t$, $t\in \mathbb{R}$, given by ${\pi }_t(v):=v(t)$. Let $\varphi :\mathbb{R}\to [1,\infty )$ be a weight function, i.e., a continuous function being non-increasing on $(-\infty ,0]$ and non-decreasing on $[0,\infty )$. Let ${\mathbf{D}}_{\varphi }$ be the subspace of $\mathbf{D}$ consisting of all $x\in \mathbf{D}$ satisfying ${\parallel v\parallel }_{\varphi }:={\parallel v\varphi \parallel }_{\infty }<\infty$ and ${lim}_{|t|\to \infty }|v(t)|=0$. The latter condition automatically holds when ${lim}_{|t|\to \infty }\varphi (t)=\infty$. We equip ${\mathbf{D}}_{\varphi }$ with the trace $\sigma$-algebra of $\mathcal{D}$, and note that this $\sigma$-algebra coincides with the $\sigma$-algebra ${\mathcal{B}}_{\varphi }^{\circ }$ on ${\mathbf{D}}_{\varphi }$ generated by the ${\parallel ·\parallel }_{\varphi }$-open balls (see Lemma 4.1 in Beutner and Zähle (2016)).

3.1. Average Value at Risk functional

Using the terminology of Part (i) of Definition A1, we obtain the following result.

Proposition 1.

Let $F\in {\mathbf{F}}_1$ and assume that F takes the value α only once. Let $\mathcal{S}$ be the set of all sequences $(G_n)\subseteq {\mathbf{F}}_1$ with $G_n\to F$ pointwise. Moreover, assume that $\int 1/\varphi (x)dx<\infty$. Then, the map ${\mathcal{R}}_{\alpha }:{\mathbf{F}}_1(\subseteq \mathbf{D})\to \mathbb{R}$ is uniformly quasi-Hadamard differentiable with respect to $\mathcal{S}$ tangentially to ${\mathbf{D}}_{\varphi }\langle {\mathbf{D}}_{\varphi }\rangle$, and the uniform quasi-Hadamard derivative ${\dot{\mathcal{R}}}_{\alpha ;F}:{\mathbf{D}}_{\varphi }\to \mathbb{R}$ is given by

$${\dot{\mathcal{R}}}_{\alpha ;F}(v):=-\int g_{\alpha }^{\prime }(F(x))v(x)dx,$$

(13)

where as before $g_{\alpha }^{\prime }:=\frac{1}{1-\alpha }{𝟙}_{(\alpha ,1]}$.

Proposition 1 shows in particular that for any $F\in {\mathbf{F}}_1$ which takes the value $\alpha$ only once, the map ${\mathcal{R}}_{\alpha }:{\mathbf{F}}_1(\subseteq \mathbf{D})\to \mathbb{R}$ is uniformly quasi-Hadamard differentiable at F tangentially to ${\mathbf{D}}_{\varphi }\langle {\mathbf{D}}_{\varphi }\rangle$ (in the sense of Part (ii) of Definition A1) with uniform quasi-Hadamard derivative given by Equation (13).

Proof. (of Proposition 1)

First, note that the map ${\dot{\mathcal{R}}}_{\alpha ;F}$ defined in Equation (13) is continuous with respect to ${\parallel ·\parallel }_{\varphi }$, because

$$|{\dot{\mathcal{R}}}_{\alpha ;F}(v_1)-{\dot{\mathcal{R}}}_{\alpha ;F}(v_2)|\le \int \frac{1}{1-\alpha }|v_1(x)-v_2(x)|dx\le \left(\frac{1}{1-\alpha }\int 1/\varphi (x)dx\right){\parallel v_1-v_2\parallel }_{\varphi }$$

holds for every $v_1,v_2\in {\mathbf{D}}_{\varphi }$.

Now, let $((F_n),v,(v_n),({\epsilon }_n))$ be a quadruple with $(F_n)\subseteq {\mathbf{F}}_1$ satisfying $F_n\to F$ pointwise, $v\in {\mathbf{D}}_{\varphi }$, $(v_n)\subseteq {\mathbf{D}}_{\varphi }$ satisfying $\parallel v_n{-v\parallel }_{\varphi }\to 0$ and $(F_n+{\epsilon }_n{v}_n)\subseteq {\mathbf{F}}_1$, and $({\epsilon }_n)\subseteq (0,\infty )$ satisfying ${\epsilon }_n\to 0$. It remains to show that

$$\underset{n\to \infty }{lim}|\frac{{\mathcal{R}}_{\alpha }(F_n+{\epsilon }_n{v}_n)-{\mathcal{R}}_{\alpha }(F_n)}{{\epsilon }_n}-{\dot{\mathcal{R}}}_{\alpha ;F}(v)|=0,$$

that is,

$$\underset{n\to \infty }{lim}|\int \left(\frac{g_{\alpha }\left(F_n(x)\right)-g_{\alpha }\left((F_n+{\epsilon }_n{v}_n)(x)\right)}{{\epsilon }_n}-\left(-g_{\alpha }^{\prime }(F(x))v(x)\right)\right)dx|=0.$$

(14)

Let us denote the integrand of the integral in Equation (14) by $I_n(x)$. In virtue of $F_n\to F$ pointwise, $\parallel v_n{-v\parallel }_{\varphi }\to 0$, ${\epsilon }_n\to 0$, and

$$|(F_n+{\epsilon }_n{v}_n)(x)-F(x)|\le |F_n(x)-F(x)|+{\epsilon }_n|v_n(x)-v(x)|+{\epsilon }_n|v(x)|,$$

we have ${lim}_{n\to \infty }{F}_n(x)=F(x)$ and ${lim}_{n\to \infty }(F_n(x)+{\epsilon }_n{v}_n(x))=F(x)$ for every $x\in \mathbb{R}$. Thus, for every $x\in \mathbb{R}$ with $F(x)<alpha$, we obtain $g_{\alpha }^{\prime }(F(x))v(x)=0$ and

$$\frac{g_{\alpha }\left(F_n(x)\right)-g_{\alpha }\left((F_n+{\epsilon }_n{v}_n)(x)\right)}{{\epsilon }_n}=0\mathrm{for}\mathrm{sufficiently}\mathrm{large}n,$$

i.e., ${lim}_{n\to \infty }{I}_n(x)=0$. Moreover, for every $x\in \mathbb{R}$ with $F(x)>\alpha$, we obtain $g_{\alpha }^{\prime }(F(x))v(x)=\frac{1}{1-\alpha }v(x)$ and

$$\frac{g_{\alpha }\left(F_n(x)\right)-g_{\alpha }\left((F_n+{\epsilon }_n{v}_n)(x)\right)}{{\epsilon }_n}=-\frac{v_n(x)}{1-\alpha }\mathrm{for}\mathrm{sufficiently}\mathrm{large}n,$$

i.e., ${lim}_{n\to \infty }{I}_n(x)=0$. Since we assumed that F takes the value $\alpha$ only once, we can conclude that ${lim}_{n\to \infty }{I}_n(x)=0$ for Lebesgue-a.e. $x\in \mathbb{R}$. Moreover, by the Lipschitz continuity of $g_{\alpha }$ with Lipschitz constant $\frac{1}{1-\alpha }$ we have

$$\begin{array}{ccc}\hfill |I_n(x)|& =& |I_n{(x)|\varphi (x)\varphi (x)}^{-1}\hfill \\ & =& |\frac{g_{\alpha }\left(F_n(x)\right)-g_{\alpha }\left((F_n+{\epsilon }_n{v}_n)(x)\right)}{{\epsilon }_n}+g_{\alpha }^{\prime }(F(x))v(x)|\varphi (x)\varphi {(x)}^{-1}\hfill \\ & \le & \frac{1}{1-\alpha }\left(\parallel v_n{\parallel }_{\varphi }+{\parallel v\parallel }_{\varphi }\right)\varphi {(x)}^{-1}\hfill \\ & \le & \frac{1}{1-\alpha }\left(\underset{n\in \mathbb{N}}{sup}\parallel v_n{\parallel }_{\varphi }+{\parallel v\parallel }_{\varphi }\right)\varphi {(x)}^{-1}.\hfill \end{array}$$

Since ${sup}_{n\in \mathbb{N}}{\parallel v_n\parallel }_{\varphi }<\infty$ (recall $\parallel v_n{-v\parallel }_{\varphi }\to 0$), the assumption $\int 1/\varphi (x)dx<\infty$ ensures that the latter expression provides a Borel measurable majorant of $I_n$. Now, the Dominated Convergence theorem implies Equation (14). ☐

As an immediate consequence of Corollary A4, Examples A1 and A2, and Proposition 1, we obtain the following corollary.

Corollary 1.

Let F, ${\widehat{F}}_n$, ${\widehat{F}}_n^{\ast }$, ${\widehat{C}}_n$, and $B_F$ be as in Example A1 (S1. or S2.) or as in Example A2 respectively, and assume that the assumptions discussed in Example A1 or in Example A2 respectively are fulfilled for some weight function ϕ with $\int 1/\varphi (x)dx<\infty$ (in particular $F\in {\mathbf{F}}_1$). Moreover, assume that F takes the value α only once. Then,

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n)-{\mathcal{R}}_{\alpha }(F)\right)⇝{\dot{\mathcal{R}}}_{\alpha ;F}(B_F)in(\mathbb{R},\mathcal{B}(\mathbb{R}))$$

and

$$\sqrt{n}\left({\mathcal{R}}_{\alpha }({\widehat{F}}_n^{\ast }(\omega ,·))-{\mathcal{R}}_{\alpha }({\widehat{C}}_n(\omega ))\right)⇝{\dot{\mathcal{R}}}_{\alpha ;F}(B_F)in(\mathbb{R},\mathcal{B}(\mathbb{R})),\mathbb{P}-a.e.\omega .$$

3.2. Compound Distribution Functional

Let ${\mathcal{C}}_p:\mathbf{F}\to \mathbf{F}$ be the compound distribution functional introduced in Section 2.1. For any $\lambda \ge 0$, let the function ${\varphi }_{\lambda }:\mathbb{R}\to [1,\infty )$ be defined by ${\varphi }_{\lambda }(x):={(1+|x|)}^{\lambda }$ and denote by ${\mathbf{F}}_{{\varphi }_{\lambda }}$ the set of all distribution functions F that satisfy $\int {\varphi }_{\lambda }(x)dF(x)<\infty$. Using the terminology of Part (ii) of Definition A1, we obtain the following Proposition 2. In the proposition, the functional ${\mathcal{C}}_p$ is restricted to the domain ${\mathbf{F}}_{{\varphi }_{\lambda }}$ in order to obtain ${\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$ as the corresponding trace. The latter will be important for Corollary 3.

Proposition 2.

Let $\lambda >{\lambda }^{\prime }\ge 0$ and $F\in {\mathbf{F}}_{{\varphi }_{\lambda }}$. Assume that ${\sum }_{k=1}^{\infty }{p}_k{k}^{(1+\lambda )\vee 2}<\infty$. Then, the map ${\mathcal{C}}_p:{\mathbf{F}}_{{\varphi }_{\lambda }}(\subseteq \mathbf{D})\to \mathbf{F}(\subseteq \mathbf{D})$ is uniformly quasi-Hadamard differentiable at F tangentially to ${\mathbf{D}}_{{\varphi }_{\lambda }}\langle {\mathbf{D}}_{{\varphi }_{\lambda }}\rangle$ with trace ${\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$. Moreover, the uniform quasi-Hadamard derivative ${\dot{\mathcal{C}}}_{p;F}:{\mathbf{D}}_{{\varphi }_{\lambda }}\to {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$ is given by

$${\dot{\mathcal{C}}}_{p;F}(v)(·):=v\ast H_{p,F}(·):=\int v(·-x)d{H}_{p,F}(x),$$

(15)

where as before $H_{p,F}:={\sum }_{k=1}^{\infty }k{p}_k{F}^{\ast (k-1)}$. In particular, if $p_m=1$ for some $m\in \mathbb{N}$, then

$${\dot{\mathcal{C}}}_{p;F}(v)(·)=m\int v(·-x)d{F}^{\ast (m-1)}(x).$$

Proposition 2 extends Proposition 4.1 of Pitts (1994). Before we prove the proposition, we note that the proposition together with Corollary A4 and Examples A1 and A2 yields the following corollary.

Corollary 2.

Let F, ${\widehat{F}}_n$, ${\widehat{F}}_n^{\ast }$, ${\widehat{C}}_n$, and $B_F$ be as in Example A1 (S1. or S2.) or as in Example A2 respectively, and assume that the assumptions discussed in Example A1 or in Example A2 respectively are fulfilled for $\varphi ={\varphi }_{\lambda }$ for some $\lambda >0$. Then, for ${\lambda }^{\prime }\in [0,\lambda )$

$$\sqrt{n}\left({\mathcal{C}}_p({\widehat{F}}_n)-{\mathcal{C}}_p(F)\right){⇝}^{\circ }{\dot{\mathcal{C}}}_{p;F}(B_F)in({\mathbf{D}}_{{\varphi }_{\lambda }^{\prime }},{\mathcal{B}}_{{\varphi }_{\lambda }^{\prime }}^{\circ },\parallel ·{\parallel }_{{\varphi }_{{\lambda }^{\prime }}})$$

and

$$\sqrt{n}\left({\mathcal{C}}_p({\widehat{F}}_n^{\ast }(\omega ,·))-{\mathcal{C}}_p({\widehat{C}}_n(\omega ))\right){⇝}^{\circ }{\dot{\mathcal{C}}}_{p;F}(B_F)in({\mathbf{D}}_{{\varphi }_{\lambda }^{\prime }},{\mathcal{B}}_{{\varphi }_{\lambda }^{\prime }}^{\circ },\parallel ·{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}),\mathbb{P}-a.e.\omega .$$

To ease the exposition of the proof of Proposition 2, we first state a lemma that follows from results given in Pitts (1994). In the sequel we use $f\ast H$ to denote the function defined by $f\ast H(·):=\int v(·-x)dH(x)$ for any measurable function f and any distribution function H of a finite (not necessarily probability) Borel measure on $\mathbb{R}$ for which $f\ast H(·)$ is well defined on $\mathbb{R}$.

Lemma 1.

Let $\lambda >{\lambda }^{\prime }\ge 0$, and $(F_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$ and $(G_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$ be any sequences such that $\parallel F_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$ and $\parallel G_n{-G\parallel }_{{\varphi }_{\lambda }}\to 0$ for some $F,G\in {\mathbf{F}}_{{\varphi }_{\lambda }}$. Then, the following two assertions hold.

(i) 

There exists a constant $C_1>0$ such that for every $k,n\in \mathbb{N}$

$$\parallel {𝟙}_{[0,\infty )}-F_n^{\ast k}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le (2^{{\lambda }^{\prime }-1}\vee 1)(1+k^{{\lambda }^{\prime }\vee 1}{C}_1).$$

(ii) 

For every $v\in {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$ there exists a constant $C_2>0$ such that for every $k,\ell ,n\in \mathbb{N}$

$$\parallel v\ast (F_n^{\ast k}\ast G_n^{\ast \ell }){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le 2^{{\lambda }^{\prime }}\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)(2+{(k+\ell )}^{{\lambda }^{\prime }\vee 1}{C}_2)\right){\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}.$$

Proof. 

(i): From Equation (2.4) in Pitts (1994) we have

$$\begin{array}{c}\hfill \parallel {𝟙}_{[0,\infty )}-F_n^{\ast k}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le (2^{{\lambda }^{\prime }-1}\vee 1)\left(1+k^{{\lambda }^{\prime }\vee 1}\int {|x|}^{{\lambda }^{\prime }}d{F}_n(x)\right),\end{array}$$

so that it remains to show that ${\int |x|}^{{\lambda }^{\prime }}d{F}_n(x)$ is bounded above uniformly in $n\in \mathbb{N}$. The functions ${𝟙}_{[0,\infty )}-F_n$ and ${𝟙}_{[0,\infty )}-F$ both lie in ${\mathbf{D}}_{{\varphi }_{\lambda }}$, because $F_n,F\in {\mathbf{F}}_{{\varphi }_{\lambda }}$. Along with $\parallel F_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$, this implies ${\int |x|}^{{\lambda }^{\prime }}d{F}_n(x)\to \int {|x|}^{{\lambda }^{\prime }}dF(x)$ (see Lemma 2.1 in Pitts (1994)). Therefore, ${\int |x|}^{{\lambda }^{\prime }}d{F}_n(x)\le C_1$ for some suitable finite constant $C_1>0$ and all $n\in \mathbb{N}$.

(ii): With the help of Lemma 2.3 of Pitts (1994) (along with $\parallel F_n^{\ast k}\ast G_n^{\ast \ell }{\parallel }_{\infty }=1$), Lemma 2.4 of Pitts (1994), and Equation (2.4) in Pitts (1994), we obtain

$$\begin{array}{cc}\hfill \parallel v\ast & {\displaystyle (F_n^{\ast k}\ast G_n^{\ast \ell }){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \hfill \le & 2^{{\lambda }^{\prime }}{\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+\parallel {𝟙}_{[0,\infty )}-F_n^{\ast k}\ast G_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\right)\hfill \\ \hfill \le & 2^{{\lambda }^{\prime }}{\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+2^{{\lambda }^{\prime }}\left(\parallel {𝟙}_{[0,\infty )}-F_n^{\ast k}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel {𝟙}_{(0,\infty )-}{G}_n^{\ast \ell }\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\right)\right)\hfill \\ \hfill \le & 2^{{\lambda }^{\prime }}{\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)\left(1+k^{{\lambda }^{\prime }\vee 1}{\int |x|}^{{\lambda }^{\prime }}d{F}_n(x)+1+{\ell }^{{\lambda }^{\prime }\vee 1}\int {|x|}^{{\lambda }^{\prime }}d{G}_n(x)\right)\right).\hfill \end{array}$$

It hence remains to show that ${\int |x|}^{{\lambda }^{\prime }}d{F}_n(x)$ and ${\int |x|}^{{\lambda }^{\prime }}d{G}_n(x)$ are bounded above uniformly in $n\in \mathbb{N}$. However, this was already done in the proof of Part (i). ☐

Proof. Proof of Proposition 2.

First, note that for $G_1,G_2\in {\mathbf{F}}_{{\varphi }_{\lambda }}$, we have

$$\begin{array}{ccc}\hfill \parallel {\mathcal{C}}_p(G_1)-{\mathcal{C}}_p(G_2){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}& \le & \parallel {\mathcal{C}}_p(G_1)-{𝟙}_{[0,\infty )}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel {𝟙}_{[0,\infty )}-{\mathcal{C}}_p(G_2)\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & \le & \int {(1+|x|)}^{{\lambda }^{\prime }}d{\mathcal{C}}_p(G_1)(x)+\int {(1+|x|)}^{{\lambda }^{\prime }}d{\mathcal{C}}_p(G_2)(x)\hfill \end{array}$$

by Equation (2.1) in Pitts (1994). Moreover, according to Lemma 2.2 in Pitts (1994), we have that the integrals ${\int |x|}^{{\lambda }^{\prime }}d{\mathcal{C}}_p(F)(x)$ and ${\int |x|}^{{\lambda }^{\prime }}d{\mathcal{C}}_p(G)(x)$ are finite under the assumptions of the proposition. Hence, ${\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$ can indeed be seen as the trace.

Second, we show $(\parallel ·{\parallel }_{{\varphi }_{\lambda }},\parallel ·{\parallel }_{{\varphi }_{{\lambda }^{\prime }}})$-continuity of the map ${\dot{\mathcal{C}}}_{p;F}:{\mathbf{D}}_{{\varphi }_{\lambda }}\to {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$. To this end let $v\in {\mathbf{D}}_{{\varphi }_{\lambda }}$ and $(v_n)\subseteq {\mathbf{D}}_{{\varphi }_{\lambda }}$ such that $\parallel v_n{-v\parallel }_{{\varphi }_{\lambda }}\to 0$. For every $k\in \mathbb{N}$, we have

$$\begin{array}{c}{\displaystyle \parallel p_{k}k(v_n-v)\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel v_n-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{p}_{k}k\left(\parallel {𝟙}_{[0,\infty )}\parallel F^{\ast (k-1)}{\parallel }_{\infty }-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel F^{\ast (k-1)}\parallel }_{\infty }\right)\hfill \\ =2^{{\lambda }^{\prime }}{\parallel v_n-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{p}_{k}k\left(\parallel {𝟙}_{[0,\infty )}-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel v_n-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{p}_{k}k\left((2^{{\lambda }^{\prime }-1}\vee 1)\left(1+{(k-1)}^{{\lambda }^{\prime }\vee 1}\int {|x|}^{{\lambda }^{\prime }}dF(x)\right)+1\right),\hfill \end{array}$$

where the first and the second inequality follow from Lemma 2.3 and Equation (2.4) in Pitts (1994) respectively. Hence,

$$\begin{array}{c}{\displaystyle \parallel {\dot{\mathcal{C}}}_{p;F}(v_n)-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}={\parallel v_n\ast H_{p,F}-v\ast H_{p,F}\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel v_n-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{\displaystyle \sum _{k=1}^{\infty }}{p}_{k}k\left((2^{{\lambda }^{\prime }-1}\vee 1)\left(1+{(k-1)}^{{\lambda }^{\prime }\vee 1}\int {|x|}^{{\lambda }^{\prime }}dF(x)\right)+1\right).\hfill \end{array}$$

Now, the series converges due to the assumptions, and $\parallel v_n{-v\parallel }_{{\varphi }_{\lambda }}\to 0$ implies $\parallel v_n{-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\to 0$. Thus, $\parallel {\dot{\mathcal{C}}}_{p;F}(v_n)-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\to 0$, which proves continuity.

Third, let $((F_n),v,(v_n),({\epsilon }_n))$ be a quadruple with $(F_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$ satisfying $\parallel F_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$, $v\in {\mathbf{D}}_{{\varphi }_{\lambda }}$, $(v_n)\subseteq {\mathbf{D}}_{{\varphi }_{\lambda }}$ satisfying $\parallel v_n{-v\parallel }_{{\varphi }_{\lambda }}\to 0$ and $(F_n+{\epsilon }_n{v}_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$, and $({\epsilon }_n)\subseteq (0,\infty )$ satisfying ${\epsilon }_n\to 0$. It remains to show that

$$\underset{n\to \infty }{lim}\parallel \frac{{\mathcal{C}}_p(F_n+{\epsilon }_n{v}_n)-{\mathcal{C}}_p(F_n)}{{\epsilon }_n}-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}=0.$$

To do so, define for $k\in {\mathbb{N}}_0$ a map $H_k:\mathbf{F}\times \mathbf{F}:\to \mathbf{F}$ by

$$H_k(G_1,G_2):={\displaystyle \sum _{j=0}^{k-1}}{G}_1^{\ast (k-1-j)}\ast G_2^{\ast j}$$

with the usual convention that the sum over the empty set equals zero. We find that for every $M\in \mathbb{N}$

$$\begin{array}{c}{\displaystyle \parallel \frac{{\mathcal{C}}_p(F_n+{\epsilon }_n{v}_n)-{\mathcal{C}}_p(F_n)}{{\epsilon }_n}-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ =\parallel \frac{1}{{\epsilon }_n}\left({\displaystyle \sum _{k=0}^{\infty }}{p}_k{(F_n+{\epsilon }_n{v}_n)}^{\ast k}-{\displaystyle \sum _{k=0}^{\infty }}{p}_k{F}_n^{\ast k}\right)-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ =\parallel \frac{1}{{\epsilon }_n}\left({\displaystyle \sum _{k=1}^{\infty }}\left(p_k{(F_n+{\epsilon }_n{v}_n)}^{\ast k}-p_k{F}_n^{\ast k}\right)\right)-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ =\parallel {\displaystyle \sum _{k=1}^{\infty }}{p}_k{v}_n\ast H_k(F_n+{\epsilon }_n{v}_n,F_n)-{\dot{\mathcal{C}}}_{p;F}(v){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ \le \parallel {\displaystyle \sum _{k=M+1}^{\infty }}{p}_k{v}_n\ast H_k(F_n+{\epsilon }_n{v}_n,F_n){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+\parallel {\displaystyle \sum _{k=1}^M}{p}_k(v_n-v)\ast H_k(F_n+{\epsilon }_n{v}_n,F_n){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ +\parallel v\ast {\displaystyle \sum _{k=M+1}^{\infty }}k{p}_k{F}^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+\parallel {\displaystyle \sum _{k=1}^M}{p}_{k}v\ast H_k(F_n+{\epsilon }_n{v}_n,F_n)-k{p}_{k}v\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ =:S_1(n,M)+S_2(n,M)+S_3(M)+S_4(n,M),\hfill \end{array}$$

where for the third “=” we use the fact that for $G_1,G_2\in \mathbf{F}$

$$(G_1-G_2)\ast H_k(G_1,G_2)=G_1^{\ast k}-G_2^{\ast k}.$$

(16)

By Part (ii) of Lemma reflemma preceding qHD of compound (this lemma can be applied since $\parallel F_n+{\epsilon }_n{v}_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$) there exists a constant $C_2>0$ such that for all $n\in \mathbb{N}$

$$\begin{array}{cc}\hfill S_1(n,M)& =\parallel {\displaystyle \sum _{k=M+1}^{\infty }}{p}_k{v}_n\ast H_k(F_n+{\epsilon }_n{v}_n,F_n){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & \le 2^{{\lambda }^{\prime }}{\parallel v_n\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{\displaystyle \sum _{k=M+1}^{\infty }}{p}_{k}k\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)\left(2+{(k-1)}^{{\lambda }^{\prime }\vee 1}{C}_2\right)\right).\hfill \end{array}$$

(17)

Since ${\lambda }^{\prime }<\lambda$ and $\parallel v_n{-v\parallel }_{{\varphi }_{\lambda }}\to 0$, we have $\parallel v_n{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le K_1$ for some finite constant $K_1>0$ and all $n\in \mathbb{N}$. Hence, the right-hand side of Equation (17) can be made arbitrarily small by choosing M large enough. That is, $S_1(n,M)$ can be made arbitrarily small uniformly in $n\in \mathbb{N}$ by choosing M large enough.

Furthermore, it is demonstrated in the proof of Proposition 4.1 of Pitts (1994) that $S_3(M)$ can be made arbitrarily small by choosing M large enough.

Next, applying again Part (ii) of Lemma 1, we obtain

$$\begin{array}{ccc}\hfill S_2(n,M)& =& \parallel {\displaystyle \sum _{k=1}^M}{p}_k(v_n-v)\ast H_k(F_n+{\epsilon }_n{v}_n,F_n){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & \le & 2^{{\lambda }^{\prime }}{\displaystyle \sum _{k=1}^M}{p}_{k}k{\parallel v_n-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)\left(2+{(k-1)}^{{\lambda }^{\prime }\vee 1}{C}_2\right)\right).\hfill \end{array}$$

Using $\parallel v_n{-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le {\parallel v_n-v\parallel }_{{\varphi }_{\lambda }}\to 0$, this term tends to zero as $n\to \infty$ for a given M.

It remains to consider the summand

$$\begin{array}{ccc}\hfill S_4(n,M)& =& \parallel {\displaystyle \sum _{k=1}^M}{p}_{k}v\ast H_k(F_n+{\epsilon }_n{v}_n,F_n)-k{p}_{k}v\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & =& \parallel {\displaystyle \sum _{k=1}^M}{p}_k{\displaystyle \sum _{\ell =0}^{k-1}}\left(v\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-v\ast F^{\ast (k-1)}\right){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}.\hfill \end{array}$$

We show that for M fixed this term can be made arbitrarily small by letting $n\to \infty$. This would follow if for every given $k\in \{1,\dots ,M\}$ and $\ell \in \{0,\dots ,k-1\}$ the expression

$$\parallel v\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-v\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$$

could be made arbitrarily small by letting $n\to \infty$. For every such k and ℓ we can find a linear combination of indicator functions of the form ${𝟙}_{[a,b)}$, $-\infty <a<b<\infty$, which we denote by $\tilde{v}$, such that

$$\begin{array}{c}{\displaystyle \parallel v\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-v\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \le \parallel v\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-\tilde{v}\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ +\parallel \tilde{v}\ast {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-\tilde{v}\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ +\parallel \tilde{v}\ast F^{\ast (k-1)}-v\ast F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ +c({\lambda }^{\prime },\tilde{v}){\parallel {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-F^{\ast (k-1)}\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ +2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \end{array}$$

(18)

for some suitable finite constant $c({\lambda }^{\prime },\tilde{v})>0$ depending only on ${\lambda }^{\prime }$ and $\tilde{v}$. The first inequality in Equation (18) is obvious (and holds for any $\tilde{v}\in {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$). The second inequality in Equation (18) is obtained by applying Lemma 2.3 of Pitts (1994) to the first summand (noting that $\parallel {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{\infty }=1$; recall $F_n+{\epsilon }_n{v}_n\in \mathbf{F}$), by applying Lemma 4.3 of Pitts (1994) to the second summand (which requires that $\tilde{v}$ is as described above), and by applying Lemma 2.3 of Pitts (1994) to the third summand.

We now consider the three summands on the right-hand side of Equation (18) separately. We start with the third term. Since $v\in {\mathbf{D}}_{{\varphi }_{\lambda }}$, Lemma 4.2 of Pitts (1994) ensures that we may assume that $\tilde{v}$ is chosen such that $\parallel \tilde{v}{-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ is arbitrarily small. Hence, for fixed M the third summand in Equation (18) can be made arbitrarily small.

We next consider the the second summand in Equation (18). Obviously,

$$\begin{array}{c}{\displaystyle \parallel {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ =\parallel {(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }-F_n^{\ast (k-1)}+F_n^{\ast (k-1)}-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ \le \parallel \left({(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}-F_n^{\ast (k-1-\ell )}\right)\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel F_n^{\ast (k-1)}-F^{\ast (k-1)}\parallel }_{{\varphi }_{{\lambda }^{\prime }}}.\hfill \end{array}$$

(19)

We start by considering the first summand in Equation (19). In view of Equation (16), it can be written as

$$\begin{array}{c}{\displaystyle \parallel \left({(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}-F_n^{\ast (k-1-\ell )}\right)\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ =\parallel \left((F_n+{\epsilon }_n{v}_n-F_n)\ast H_{k-1-\ell }(F_n+{\epsilon }_n{v}_n,F_n)\right)\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ =\parallel \left({\epsilon }_n{v}_n\ast H_{k-1-\ell }(F_n+{\epsilon }_n{v}_n,F_n)\right)\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}.\hfill \end{array}$$

Applying Lemma 2.3 of Pitts (1994) with $f={\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)$ and $H=F_n^{\ast \ell }$ we obtain

$$\begin{array}{c}{\displaystyle \parallel \left({\epsilon }_n{v}_n\ast H_{k-1-\ell }(F_n+{\epsilon }_n{v}_n,F_n)\right)\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \le 2^{{\lambda }^{\prime }}\parallel \left({\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)\right){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}\parallel F_n^{\ast \ell }{\parallel }_{\infty }-F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel F_n^{\ast \ell }\parallel }_{\infty }\right)\hfill \\ =2^{{\lambda }^{\prime }}\parallel \left({\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)\right){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ \le 2^{{\lambda }^{\prime }}\parallel \left({\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)\right){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left\{(2^{{\lambda }^{\prime }-1}\vee 1)\left(1+{\ell }^{{\lambda }^{\prime }\vee 1}{C}_1\right)+1\right\},\hfill \end{array}$$

(20)

where we applied Part (i) of Lemma 1 to $\parallel {𝟙}_{[0,\infty )}-F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ to obtain the last inequality. Hence, for the left-hand side of Equation (20) to go to zero as $n\to \infty$ it suffices to show that $\parallel ({\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\to 0$ as $n\to \infty$. The latter follows from

$$\begin{array}{c}{\displaystyle \parallel \left({\epsilon }_n{v}_n\ast H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)\right){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}}\hfill \\ \le 2^{{\lambda }^{\prime }}(k-\ell -1){\epsilon }_n{\parallel v_n\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)\left(2+{((k-\ell -2))}^{{\lambda }^{\prime }\vee 1}{C}_2\right)\right),\hfill \end{array}$$

(21)

where we applied Part (ii) of Lemma 1 with $v={\epsilon }_n{v}_n$ to all summands in $H_{k-\ell -1}(F_n+{\epsilon }_n{v}_n,F_n)$. For every k and $\ell \in \{0,\dots ,k-1\}$ this expression goes indeed to zero as $n\to \infty$, because, as mentioned before, $\parallel v_n{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ is uniformly bounded in $n\in \mathbb{N}$, and we have ${\epsilon }_n\to 0$. Next, we consider the second summand in Equation (19). Applying Equation (16) to $F_n^{\ast (k-1)}$ and $F^{\ast (k-1)}$ and subsequently Part (ii) of Lemma 1 to the summands in $H_{k-1}(F_n,F)$, we have

$$\parallel F_n^{\ast (k-1)}-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le 2^{{\lambda }^{\prime }}(k-1){\parallel F_n-F\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)(2+{((k-2))}^{{\lambda }^{\prime }\vee 1}{C}_2)\right).$$

Clearly for every k this term goes to zero 0 as $n\to \infty$, because $\parallel F_n{-F\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le {\parallel F_n-F\parallel }_{{\varphi }_{\lambda }}\to 0$ as $n\to \infty$ by assumption. This together with the fact that Equation (20) goes to zero 0 as $n\to \infty$ shows that Equation (19) goes to zero in ${\parallel ·\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ as $n\to \infty$. Therefore, the second summand in Equation (18) goes to zero as $n\to \infty$.

It remains to consider the first term in Equation (18). We find

$$\begin{array}{c}{\displaystyle 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{\lambda }}\left(\parallel {𝟙}_{[0,\infty )}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)}\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-F^{\ast (k-1)}+F^{\ast (k-1)}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel {𝟙}_{[0,\infty )}-F^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+{\parallel F^{\ast (k-1)}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right)\hfill \\ \le 2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}(2^{{\lambda }^{\prime }-1}\vee 1)\left(1+k^{\lambda \vee 1}\int {|x|}^{{\lambda }^{\prime }}dF(x)\right)\hfill \\ +2^{{\lambda }^{\prime }}{\parallel \tilde{v}-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\left(\parallel F^{\ast (k-1)}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}+1\right),\hfill \end{array}$$

(22)

where for the last inequality we used Formula (2.4) of Pitts (1994). In the following, Equation (19) we showed that $\parallel F^{\ast (k-1)}-{(F_n+{\epsilon }_n{v}_n)}^{\ast (k-1-\ell )}\ast F_n^{\ast \ell }{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ goes to zero as $n\to \infty$ for every k and $\ell \in \{0,\dots ,k-1\}$. Hence, for every such k and ℓ, it is uniformly bounded in $n\in \mathbb{N}$. Therefore, we can make Equation (22) arbitrarily small by making $\parallel \tilde{v}{-v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}$ small which, as mentioned above, is possible according to Lemma 4.2 of Pitts (1994). This finishes the proof. ☐

3.3. Composition of Average Value at Risk Functional and Compound Distribution Functional

Here, we consider the composition of the Average Value at Risk functional ${\mathcal{R}}_{\alpha }$ defined in Equation (1) and the compound distribution functional ${\mathcal{C}}_p$ introduced in Section 2.1. As a consequence of Propositions 1 and 2, we obtain the following Corollary 3. Note that, for any $\lambda >1$, Lemma 2.2 in Pitts (1994) yields ${\mathcal{C}}_p({\mathbf{F}}_{{\varphi }_{\lambda }})\subseteq {\mathbf{F}}_1$ so that the composition ${\mathcal{R}}_{\alpha }\circ {\mathcal{C}}_p$ is well defined on ${\mathbf{F}}_{{\varphi }_{\lambda }}$.

Corollary 3.

Let $\lambda >1$ and assume ${\sum }_{k=1}^{\infty }{p}_k{k}^{1+\lambda }<\infty$. Let $F\in {\mathbf{F}}_{{\varphi }_{\lambda }}$, and assume that ${\mathcal{C}}_p(F)$ takes the value α only once. Then, the map $T_{\alpha ,p}:={\mathcal{R}}_{\alpha }\circ {\mathcal{C}}_p:{\mathbf{F}}_{{\varphi }_{\lambda }}(\subseteq \mathbf{D})\to \mathbb{R}$ is uniformly quasi-Hadamard differentiable at F tangentially to ${\mathbf{D}}_{{\varphi }_{\lambda }}\langle {\mathbf{D}}_{{\varphi }_{\lambda }}\rangle$, and the uniform quasi-Hadamard derivative ${\dot{T}}_{\alpha ,p;F}:{\mathbf{D}}_{{\varphi }_{\lambda }}\to \mathbb{R}$ is given by ${\dot{T}}_{\alpha ,p;F}={\dot{\mathcal{R}}}_{\alpha ;{\mathcal{C}}_p(F)}\circ {\dot{\mathcal{C}}}_{p;F}$, i.e.,

$${\dot{T}}_{\alpha ,p;F}(v)=\int g_{\alpha }^{\prime }({\mathcal{C}}_p(F)(x))(v\ast H_{p,F})(x)dxforallv\in {\mathbf{D}}_{{\varphi }_{\lambda }}$$

with $g_{\alpha }^{\prime }$ and $v\ast H_{p,F}$ as in Proposition 1 and 2, respectively.

Proof. 

We intend to apply Lemma A1 to $H={\mathcal{C}}_p:{\mathbf{F}}_{{\varphi }_{\lambda }}\to {\mathbf{F}}_1$ and $\tilde{H}={\mathcal{R}}_{\alpha }:{\mathbf{F}}_1\to \mathbb{R}$. To verify that the assumptions of the lemma are fulfilled, we first recall from the comment directly before Corollary 3 that ${\mathcal{C}}_p({\mathbf{F}}_{{\varphi }_{\lambda }})\subseteq {\mathbf{F}}_1$. It remains to show that the Assumptions (a)–(c) of Lemma A1 are fulfilled. According to Proposition 2 we have that for every ${\lambda }^{\prime }\in (1,\lambda )$ the functional ${\mathcal{C}}_p$ is uniformly quasi-Hadamard differentiable at F tangentially to ${\mathbf{D}}_{{\varphi }_{\lambda }}\langle {\mathbf{D}}_{{\varphi }_{\lambda }}\rangle$ with trace ${\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$, which is the first part of Assumption (b). The second part of Assumption (b) means ${\dot{\mathcal{C}}}_{p,F}({\mathbf{D}}_{{\varphi }_{\lambda }})\subseteq {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}$ and follows from

$$\begin{array}{ccc}\hfill \parallel {\dot{\mathcal{C}}}_{p;F}{(v)\parallel }_{{\varphi }_{{\lambda }^{\prime }}}& =& \parallel v\ast {\displaystyle \sum _{k=1}^{\infty }}{p}_{k}k{F}^{\ast (k-1)}{\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & \le & 2^{{\lambda }^{\prime }}{\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{\displaystyle \sum _{k=1}^{\infty }}{p}_{k}k\left(1+(2^{{\lambda }^{\prime }-1}\vee 1)\left(1+k^{{\lambda }^{\prime }\vee 1}\int {|x|}^{{\lambda }^{\prime }}dF(x)\right)\right)\hfill \end{array}$$

(for which we applied Lemma 2.3 and Inequality (2.4) in Pitts (1994)), the convergence of the latter series (which holds by assumption), and ${\parallel v\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\le {\parallel v\parallel }_{{\varphi }_{\lambda }}<\infty$. Further, it follows from Proposition 1 that the map ${\mathcal{R}}_{\alpha }$ is uniformly quasi-Hadamard differentiable tangentially to ${\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}\langle {\mathbf{D}}_{{\varphi }_{{\lambda }^{\prime }}}\rangle$ at every distribution function of ${\mathbf{F}}_{{\varphi }_{{\lambda }^{\prime }}}$ that takes the value $1-\alpha$ only once. This is Assumption (c) of Lemma A1.

It remains to show that Assumption (a) of Lemma A1 also holds true. In the present setting, Assumption (a) means that for every sequence $(F_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$ with $\parallel F_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$ we have ${\mathcal{C}}_p(F_n)\to {\mathcal{C}}_p(F)$ pointwise. We show that we even have $\parallel {\mathcal{C}}_p(F_n)-{\mathcal{C}}_p(F){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\to 0$. Thus, let $(F_n)\subseteq {\mathbf{F}}_{{\varphi }_{\lambda }}$. Then,

$$\begin{array}{ccc}\hfill \parallel {\mathcal{C}}_p(F_n)-{\mathcal{C}}_p(F){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}& =& \parallel {\displaystyle \sum _{k=1}^{\infty }}{p}_k(F_n^{\ast k}-F^{\ast k}){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & =& \parallel (F_n-F)\ast {\displaystyle \sum _{k=1}^{\infty }}{p}_k{H}_k(F_n,F){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\hfill \\ & \le & 2^{{\lambda }^{\prime }}{\parallel F_n-F\parallel }_{{\varphi }_{{\lambda }^{\prime }}}{\displaystyle \sum _{k=1}^{\infty }}{p}_{k}k\left(1+2^{{\lambda }^{\prime }}(2^{{\lambda }^{\prime }-1}\vee 1)\left(2+{(k-1)}^{{\lambda }^{\prime }\vee 1}{C}_2\right)\right),\hfill \end{array}$$

where we used Equation (16) for the second “=” and applied Part (ii) of Lemma 1 to the summands of $H_k$ to obtain the latter inequality. Since the series converges, we obtain $\parallel {\mathcal{C}}_p(F_n)-{\mathcal{C}}_p(F){\parallel }_{{\varphi }_{{\lambda }^{\prime }}}\to 0$ when assuming $\parallel F_n{-F\parallel }_{{\varphi }_{\lambda }}\to 0$. ☐

As an immediate consequence of Corollary A4, Examples A1 and A2, and Corollary 3, we obtain the following corollary.

Corollary 4.

Let F, ${\widehat{F}}_n$, ${\widehat{F}}_n^{\ast }$, ${\widehat{C}}_n$, and $B_F$ be as in Example A1 (S1. or S2.) or as in Example A2, respectively, and assume that the assumptions discussed in Example A1 or in Example A2 respectively are fulfilled for $\varphi ={\varphi }_{\lambda }$ for some $\lambda >1$ (in particular $F\in {\mathbf{F}}_1$). Moreover, assume ${\sum }_{k=1}^{\infty }{p}_k{k}^{1+\lambda }<\infty$ and that ${\mathcal{C}}_p(F)$ takes the value α only once. Then,

$$\sqrt{n}\left(T_{\alpha ,p}({\widehat{F}}_n)-T_{\alpha ,p}(F)\right)⇝{\dot{T}}_{\alpha ,p;F}(B_F)in(\mathbb{R},\mathcal{B}(\mathbb{R}))$$

and

$$\sqrt{n}\left(T_{\alpha ,p}({\widehat{F}}_n^{\ast }(\omega ,·))-T_{\alpha ,p}({\widehat{C}}_n(\omega ))\right)⇝{\dot{T}}_{\alpha ,p;F}(B_F)in(\mathbb{R},\mathcal{B}(\mathbb{R})),\mathbb{P}-a.e.\omega .$$

4. Conclusions

In this paper, we consider the sub-additive risk measure Average Value at Risk and presented in Section 2.1 and Section 2.2 results on almost sure bootstrap consistency for the corresponding empirical plug-in estimator based on i.i.d. or strictly stationary, geometrically $\beta$-mixing observations. Our results supplement those by Beutner and Zähle (2016) on bootstrap consistency in probability and those by Sun and Cheng (2018) on bootstrap consistency in probability for the Tail Conditional Expectation (which is not sub-additive). In Section 2.1, we also look at the case where one is interested in Average Value of Risk in the collective risk model. Note that one might interpret the collective risk model as a pooling of independent risks. In the context of Solvency II, pooling of risks has received increased attention (see, for example, Bølviken and Guillen 2017). However, one should keep in mind that our results of Section 2.1 can typically not be applied in the Solvency II context. In Solvency II applications risks are usually dependent, whereas in the collective risk model the different risks (claims) are assumed to be independent.