Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications

Uniform asymptotics for the tail probability of weighted sums with heavy tails

This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function, the uniform asymptotic equivalence of the tail probabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$ and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent and follow the long-tailed distribution, and $c_i$'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of $X_i$'s w…

Convolutions of long-tailed and subexponential distributions

Convolutions of long-tailed and subexponential distributions play a major role in the analysis of many stochastic systems. We study these convolutions, proving some important new results through a simple and coherent approach, and showing also that the standard properties of such convolutions follow as easy consequences.

Tail Behavior of Randomly Weighted Sums

Let $\{X_t, t \geq 1\}$ be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails and $\{ Θ_t, t\geq1 \}$ be a sequence of positive random variables independent of the sequence $\{X_t, t \geq 1\}$. We shall discuss the tail probabilities and almost sure convergence of $\xinf=\sum_{t=1}^{\infty}Θ_t X_t^{+}$ (where $X^+=\max\{0,X\}$) and $\max_{1\leq k<\infty} \sum_{t=1}^{k}Θ_t X_t$ and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illu…

Moments and tails of hitting times of Bessel processes and convolutions of elementary mixtures of exponential distributions

We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.

Asymptotic distributions for weighted power sums of extreme values

Let $X_{1,n}\le\cdots\le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail with tail index $γ$. Given known constants $d_{i,n}$, $1\le i\le n$, consider the weighted power sums $\sum^{k_n}_{i=1}d_{n+1-i,n}\log^pX_{n+1-i,n}$, where $p>0$ and the $k_n$ are positive integers such that $k_n\to\infty$ and $k_n/n\to0$ as $n\to\infty$. Under some constraints on the weights $d_{i,n}$, we prove asymptotic normality for the power sums over the whole heavy-tail model. We apply the obtained result to construct a new class o…

Characterization of tails through hazard rate and convolution closure properties

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.

A New Information Theoretical Concept: Information-Weighted Heavy-tailed Distributions

Given an arbitrary continuous probability density function, it is introduced a conjugated probability density, which is defined through the Shannon information associated with its cumulative distribution function. These new densities are computed from a number of standard distributions, including uniform, normal, exponential, Pareto, logistic, Kumaraswamy, Rayleigh, Cauchy, Weibull, and Maxwell-Boltzmann. The case of joint information-weighted probability distribution is assessed. An additive property is derived in the case of independent variables. One-sided and two-sided information-weightin…

Asymptotics of convolution with the semi-regular-variation tail and its application to risk

In this paper, according to a certain criterion, we divide the exponential distribution class into three subclasses. One of them is closely related to the regular-variation-tailed distribution class, so it is called the semi-regular-variation-tailed distribution class. In the class, although all distributions are not convolution equivalent,they still have some good properties. We give the precise tail asymptotic expression of convolution of these distributions, and prove that the new class is closed under convolution. In addition, we do not need to require the corresponding random variables to…

Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks

Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ,X_{n}$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and $% X_{1},\ldots ,X_{n}$ may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory.

高阶条件尾部矩对于卷积等价分布损失的渐近性

Asymptotics of higher-order conditional tail moments for convolution-equivalently distributed losses

本文研究系统性崩溃下高阶条件尾部矩的渐近性，针对两个投资组合的相依损失，分别处理轻尾卷积等价分布和重尾正则变化分布，证明渐近行为对相依强度稳健，并通过数值模拟验证。

Density and tails of unimodal convolution semigroups

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.

On Beta-Product Convolutions

Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of R and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. Our first application concerns the asymptotic behaviour of the componentwise sample minima related to an elliptical distributions. Further, we derive the lower tails asymptotic of the aggregated risk for bivariate polar distributions.

On the long tail property of product convolution

Let $X$ and $Y$ be two independent random variables with corresponding distributions $F$ and $G$ supported on $[0,\infty)$. The distribution of the product $XY$, which is called the product convolution of $F$ and $G$, is denoted by $H$. In this paper, some suitable conditions about $F $ and $G $ are given, under which the distribution $H$ belongs to the long-tailed distribution class. Here, $F$ is a generalized long-tailed distribution and is not necessarily an exponential distribution. Finally, a series of examples are given to show that the above conditions are satisfied by many distribution…

Berry-Esseen and Edgeworth approximations for the tail of an infinite sum of weighted gamma random variables

Consider the sum $Z = \sum_{n=1}^\infty λ_n (η_n - \mathbb{E}η_n)$, where $η_n$ are i.i.d.~gamma random variables with shape parameter $r > 0$, and the $λ_n$'s are predetermined weights. We study the asymptotic behavior of the tail $\sum_{n=M}^\infty λ_n (η_n - \mathbb{E}η_n)$ which is asymptotically normal under certain conditions. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. We illustrate the effectiveness of these expansions on an infinite sum of weighted chi-squared distributions.

Lower limits and equivalences for convolution tails

Suppose $F$ is a distribution on the half-line $[0,\infty)$. We study the limits of the ratios of tails $\bar{F*F}(x)/\bar{F}(x)$ as $x\to\infty$. We also discuss the classes of distributions ${\mathcal{S}}$, ${\mathcal{S}}(γ)$ and ${\mathcal{S}}^*$.

A phase transition for tails of the free multiplicative convolution powers

We study the behavior of the tail of a measure $μ^{\boxtimes t}$, where $\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq 1$. We focus on the case where $μ$ is a probability measure on the positive half-line with a regularly varying tail i.e. of the form $x^{-α} L(x)$, where $L$ is slowly varying. We obtain a phase transition in the behavior of the tail of $μ^{\boxplus t}$ between regimes $α<1$ and $α>1$. Our main tool is a description of the regularly varying tails of $μ$ in terms of the behavior of the corresponding $S$-transform at $0^-$. We also describe the ta…

Tail asymptotics for exponential functionals of Levy processes: the convolution equivalent case

We determine the rate of decrease of the right tail distribution of the exponential functional of a Levy process with a convolution equivalent Levy measure. Our main result establishes that it decreases as the right tail of the image under the exponential function of the Levy measure of the underlying Levy process. The method of proof relies on fluctuation theory of Levy processes and an explicit path-wise representation of the exponential functional as the exponential functional of a bivariate subordinator. Our techniques allow us to establish rather general estimates of the measure of the ex…

广义拟渐近独立下随机加权和的联合尾部

Joint tail of randomly weighted sums under generalized quasi asymptotic independence

本文在广义拟渐近独立下研究二维随机加权和的联合尾部，提出新依赖结构与二维重尾类结合的方法，并给出有限时间破产概率。

Large deviations for infinite weighted sums of stretched exponential random variables

We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtmüller (2000), who study the same objects under the assumption of finite exponential moments, and Gantert et al.\ (2014), who study finite weighted sums with stretched exponential tails.

关于重尾卷积半群上的单调加法统计

Monotone additive statistics on heavy-tailed convolution semigroups

研究概率测度半群的子半群及其中的单调加法统计，证明在有限p阶矩条件下，期望的标量倍数是唯一的单调加法统计，且整个概率测度半群无非零单调加法统计。

具有相互相依求和项的随机加权和的尾部行为

Tail behavior of randomly weighted sums with interdependent summands

本文研究相互相依随机变量及其随机权重的加权和尾部行为，证明最大和等价性，并给出离散时间风险模型中的有限时与随机时破产概率及广义矩渐近界。

Tail Asymptotic Behavior of the supremum of a class of chi-square processes

In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively.

Weighted Bounded Mean Oscillation applied to Backward Stochastic Differential Equations

We deduce conditional $L_p$-estimates for the variation of a solution of a BSDE. Both quadratic and sub-quadratic types of BSDEs are considered, and using the theory of weighted bounded mean oscillation we deduce new tail estimates for the solution $(Y,Z)$ on subintervals of $[0,T]$. Some new results for the decoupling technique introduced in \cite{jossain} are obtained as well and some applications of the tail estimates are given.

Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $η$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution function of the random sum $S_η=ξ_1+ξ_2+\cdots+ξ_η$ belongs to the class of $\mathcal{O}$-exponential distributions.

随机加权和的渐进行为无需随机权重的矩条件

Asymptotics of randomly weighted sums without moment conditions of random weights

研究在无矩条件下随机加权和的渐进行为，应用于离散时间风险模型中有限时间破产概率的估计，特别处理具有常规变化尾部的增量情况。

黎曼流形上随机游走导出的加权张量和函数的尾部界

Tail Bounds for Functions of Weighted Tensor Sums Derived from Random Walks on Riemannian Manifolds

本文利用Mond-Pecaric方法建立新张量不等式，并结合Laplace-Beltrami算子谱信息，为流形上随机游走导出的加权张量和给出新的尾部界及转移矩阵列和估计。

加权样本协方差的渐近谱：谱收敛的另一种证明

Asymptotic spectrum of weighted sample covariance: another proof of spectrum convergence

本文为加权样本协方差的高维谱收敛提供了一个更简洁自足的替代证明，并分析了不同权重分布及重尾情形下有限样本的谱行为。

On the convex infimum convolution inequality with optimal cost function

We show that every symmetric random variable with log-concave tails satisfies the convex infimum convolution inequality with an optimal cost function (up to scaling). As a result, we obtain nearly optimal comparison of weak and strong moments for symmetric random vectors with independent coordinates with log-concave tails.

On closedness under convolution roots related to an infinitely divisible distribution in the distribution class L(γ)

We consider questions related to the well-known conjecture due to Embrechts and Goldie on the closedness of different classes of heavy- and light-tailed distributions with respect to convolution roots. We show that the class L(γ)\cap OS is not closed under convolution roots related to an infinitely divisible distribution for any γ\ge0, i.e. we provide examples of infinitely divisible distributions belonging to this class such that the corresponding Levy spectral distribution does not. We also prove a similar statement for the class (L(γ)\cap OS)\ S(γ). In order to facilitate our analysis, we e…

On a transformation between distributions obeying the principle of a single big jump

Beck et al. (2013) introduced a new distribution class J which contains many heavy-tailed and light-tailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavy-tailed distributions to light-tailed ones, we find some light-tailed distributions, which belong to the class J but do not belong to the convolution equivalent distribution class and which are not even weakly tail equivalent to any convolution equivalent distribution. This fact helps to understand the structure of the light-tailed distributions in the class J and leads to a negative an…

二维随机加权和尾部概率的二阶渐近分析

Second-order Asymptotic Analysis of Tail Probabilities of Bidimensional Randomly Weighted Sums

研究保险二维离散时间风险模型中总净损失随机折现价值两类尾部概率的二阶渐近性，通过建立随机加权和公式并应用于模型，得到二阶渐近公式，数值模拟显示其比一阶更精确。

加权随机连接模型中固定度节点的泊松近似

Poisson approximation of fixed-degree nodes in weighted random connection models

本文研究了高密度加权随机连接模型中固定度顶点的泊松近似问题，基于权重分布左尾的小权重分位数建立了一般性准则，并通过截断论证与Poisson点过程泛函的定量泊松近似结果进行证明。

Functional inequalities for heavy tails distributions and application to isoperimetry

This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures on $\R^n$ we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve …

Large Deviations for Weighted Sums of Stretched Exponential Random Variables

We consider the probability that a weighted sum of $n$ i.i.d. random variables $X_j$, $j = 1, . . ., n$, with stretched exponential tails is larger than its expectation and determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller [8] as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev [10, 11]. As an application of our result, motivated by…

一类重尾分布的风险聚合与随机占优

Risk aggregation and stochastic dominance for a class of heavy-tailed distributions

本研究提出一类新的重尾分布，证明其独立同分布变量的加权平均在随机序下占优于单变量，且该结论可推广至负相关、非同分布情形，后者的加权平均还占优于分布混合，为风险聚合分析提供了理论支撑。

独立双边指数随机变量和的尾部界

Tail bounds for sums of independent two-sided exponential random variables

本文为双边指数随机变量加权和建立匹配主项的尾部上下界，推广了Janson的单边指数结果。

在具有厚尾风险模型中恢复收敛性：一种用于稳健回测的加权 Kolmogorov 方法

A Weighted Kolmogorov Metric under Sub-Cubic Moments: Admissible Parameters and an Obstruction

本文提出了一种加权 Kolmogorov 度量，用于解决高频率金融数据中厚尾特性导致的收敛问题，通过引入耗尽函数和截断方案，恢复了最优的高斯收敛速率。

二维重尾分布的一种新方法

A new approach in two-dimensional heavy-tailed distributions

本文提出定义二维重尾分布的新方法，引入三类二维重尾分布并研究其闭包性质，推广至随机加权和，并应用于二维风险模型的破产概率渐近表达。