Pawel Hitczenko
We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the non--negative case our bound is similar to a lower bound obtained by Goldie and Grübel in 1996.
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On the tails of Dickman-like perpetuities
通过概率技术研究接近Dickman分布的永续年金尾部渐进行为
Consider a Lamperti-Kiu Markov additive process $(J_t,ξ_t:t\geq0)$ on $\{+,-\}\times\mathbb{R}\cup\infty$ where $J$ is the modulating Markov chain component. First, we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramer's condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.
We consider the random variables $R$ which are solutions of the distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent of $R$ and $\ABS{M}\leq 1$. Goldie and Grübel showed that the tails of $R$ are no heavier than exponential. In this note we provide the exact lower and upper bounds of the domain of the Laplace transform of $R$.
An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient conditions which ensure that the distribution right tail of a perpetuity $\mathbb{P}\{X>x\}$ is asymptotic to $ax^ce^{-bx}$ as $x\to\infty$ for some $a,b>0$ and $c\in\mathbb{R}$. Our results complement those of Denisov and Zwart [J. Appl. Probab. 44 (2007), 1031--1046]. As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential…
We consider the tail behavior of random variables $R$ which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is independent of $R$ and $|M|\le 1$. Goldie and Grübel showed that the tails of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$ resembles near 1 the uniform distribution, then the tails of $R$ are Poissonian. In this paper, we further investigate the connection between the tails of $R$ and the behavior of $M$ near 1. We focus on the special case when $Q$ is constant and $M$ is nonnegative.
On the empirical spectral distribution of matrix perpetuities
研究矩阵永续性（即形如X = A X A^T + B的仿射不动点方程）的经验谱分布，建立存在唯一性结果，分析正交不变模型下期望经验谱分布的幂律尾渐近，并证明在次临界区域弱收敛到自由永续分布。
On perpetuities with light tails
本文研究了轻尾永续过程的渐进行为，分析了依赖结构对对数尾渐进行为的影响，得出了在独立和依赖情况下 $-\log\mathbb{P}(R>x)$ 的精确渐进行为。
We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator…
Perpetuities with light tails and the local dependence measure
本文针对仿射随机不动点方程的轻尾解，引入局部依赖测度与勒让德型变换刻画其对数右尾渐近性，并拓展至相关递推序列，构造确定性标度得到几乎必然正有限的上极限显式值。
Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.]…
Free Perpetuities I: Existence, Subordination and Tail Asymptotics
本文研究自由概率中仿射不动点方程的解，涵盖次临界与临界情形，证明临界时级数几乎一致收敛但无高阶矩，并利用从属技术得到尾部幂律渐近，类比经典Kesten定理。
In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as $x\to\infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln P(M>1-\frac1x)$. Random variable $R$ satisfies the stochastic equation $R\stackrel{d}{=}1+MR$ with $M$ and $R$ independent, thus this result fits into the study of tails of iterated random equations, or more specifically, of perpetuities.
We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126--166], who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approac…
The KPZ Upper-Tail Field as a KPZ Fixed Point with Brownian Initial Data
本文通过有向景观变分和局部布朗比较，将KPZ上尾场正时间扇区识别为从双边布朗V-型轮廓出发的KPZ不动点，负时间扇区用指数倾斜表示，并恢复其大尺度极限。
Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a stationary sequence $(ξ_n:n\geq 0)$ to produce the process $(S_n+ξ_n:n\geq0)$. This paper is concerned with computing the distribution of the all-time maximum $M_{\infty}=\max \{S_k+ξ_k:k\geq0\}$ of perturbed random walk with a negative drift. Such a maximum arises in several different applications settings, including production systems, communications networks and insurance risk. Our main results describe asymptotics for $\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics depend greatly on whether the $ξ_n$'s are…
For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where w…
Monotone tail functions: definitions, properties, and application to risk-reducing strategies
本文推广了单调尾部函数的理论，刻画了尾部变换下分位数的表达方式，并将其应用于期权定价、保险精算、象限完全相依分析及经济资本风险降低策略。
Tail Asymptotics of the Signature of various stochastic processes and its connection to the Quadratic Variation
本文扩展了布朗运动签名与二次变差联系的研究，证明了半鞅Itô签名与流逝时间及分数布朗运动fWIS签名与Hurst参数的联系，并提出了多维扩展猜想。
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…
We investigate the tail behaviour of the steady state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queuing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.
Let $N$ be the number of triangles in an Erdős-Rényi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the upper tail problem for $N,$ namely, we investigate how fast $k$ must grow, so that the probability of $\{N\ge k\}$ is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by P…
We examine three methods of constructing correlated Student-$t$ random variables. Our motivation arises from simulations that utilise heavy-tailed distributions for the purposes of stress testing and economic capital calculations for financial institutions. We make several observations regarding the suitability of the three methods for this purpose.
The exact expression for the probability density $p_{_N}(x)$ for sums of a finite number $N$ of random independent terms is obtained. It is shown that the very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random terms are distributed according to the Gauss Law. In all other cases the tail for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms diverge the non-Gaussian tail is related to a Levy distribution for $p_{_N}(x)$. However, the tail is not Gaussian even if the variances are finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At sm…
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