Jean-Baptiste Bardet, Hélène Guerin, Florent Malrieu
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 $R$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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
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.
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.
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.
We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities and thus not satisfying Kesten's theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence equations and a new central limit theorem for GARCH(1,1) processes in the critical case.
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。
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.
Tail Dependence of Multivariate Archimedean Copulas
本文将多元 Archimedean copulas 的尾部相依性研究扩展至缓慢变化和快速变化的 Laplace 变换，证明了不同尾部阶数的尾部相依性函数能有效捕捉整个 Archimedean copulas 类的极端相依性及全部尾部行为谱。
In this paper, we will give a sufficient condition for a non-negative random variable $X$ to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.
Right-tail asymptotics for products of independent normal random variables
研究独立正态变量乘积的右尾概率渐进行为，提出显式渐近公式及修正项，采用边界 saddle-point/Laplace 方法证明。
The Laplace transform is a widely used tool in the study of probability distributions, often allowing for a probability density functions and distribution functions simpler determination and being a moments generating function. In this paper it is considered a situation not so simple, as it is the case of the MGinf queue busy period length distribution. Attention will also be given the respective tail Laplace transform. Then, in the context of an open queues network, which nodes behave as MGinf queues, the Laplace transform will be used to construct an algorithm to determine the Laplace transf…
This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices. This machinery is used to derive eigenvalue analogues of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonorm…
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 branching random walk (BRW) smoothing transform $T$ is defined as $T:\text{distr}(U_{1})\mapsto \text{distr} (\sum_{i=1}^{L}X_{i}U_{i})$, where given realizations $\{X_{i}\}_{i=1}^{L}$ of a point process, $U_{1},U_{2},...$ are conditionally independent identically distributed random variables, and $0\leq \text{Prob}\{L=\infty \}\leq 1$. Given $α\in (0,1]$, $α$-\emph{elementary} fixed points are fixed points of $T$ whose Laplace-Stieltjes transforms $ϕ$ satisfy $\underset{s\to +0}{\lim}\dfrac{1-ϕ(s)}{s^α}=m$, where $m$ is any given positive number. If $α=1$, these are the fixed points with …
We consider a multivariate distributional recursion of sum-type as arising in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.
We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.
In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this work extends previous work by considering the tensors summation tail behavior of the top $k$-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in Shih Yu's work: Convenient tail bounds for sums of random tensors. Majorization and antisymmetric tensor product tools are main techniques …
In this paper we provide explicit representations of Laplace transforms of extinction time and total progeny of rough continuous-state branching processes introduced in [7]. Also, we show that their tail distributions are much fatter than those of Feller branching diffusions.
We consider two independent random variables with the given tail asymptotic (e.g. power or exponential). We find tail asymptotic for their sum and product. This is done by some cumbersome but purely technical computations and requires the use of the Laplace method for asymptotic of integrals. We also recall the results for asymptotic of a self-similar locally stationary centered Gaussian process plus a deterministic drift; and we find the asymptotic for the same probability after multiplying the drift by a random variable, which is independent of this process. Keywords: tail asymptotic, Laplac…
The variance-gamma product distribution
该研究推导了零位置参数下N个独立方差-伽马随机变量乘积的精确概率密度函数，拓展得到相关分布函数与渐近近似，并进一步推出非对称拉普拉斯变量乘积及拉普拉斯与中心化正态变量混合乘积的分布闭式解。
For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to …
We use Mellin transforms to compute a full asymptotic expansion for the tail of the Laplace transform of the squared $L^2$-norm of any multiply-integrated Brownian sheet. Through reversion we obtain corresponding strong small-deviation estimates.
Risk theory in a finite customer-pool setting
针对有限主要客户与大量小型客户的保险模型，本文用谱正 Lévy 过程刻画小型客户动态，推导破产概率闭式 Laplace 变换、主要客户情形的尾部渐近及初始准备金超出量分布。
We consider a processor shared $M/M/1$ queue that can accommodate at most a finite number $K$ of customers. We give an exact expression for the sojourn time distribution in the finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit $K\to\infty$, by locating the dominant singularity of the Laplace transform.
We give a sufficient condition for the exponential decay of the tail probability of a non-negative random variable. We consider the Laplace-Stieltjes transform of the probability distribution function of the random variable. We present a theorem, according to which if the abscissa of convergence of the LS transform is negative finite and the real point on the axis of convergence is a pole of the LS transform, then the tail probability decays exponentially. For the proof of the theorem, we extend and apply so-called a finite form of Ikehara's complex Tauberian theorem by Graham-Vaaler.
We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincaré inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp-Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Th…
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in $\Z^d$. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent …
We consider the discrete Laplace operator $Δ^{(N)}$ on Erdős--Rényi random graphs with $N$ vertices and edge probability $p/N$. We are interested in the limiting spectral properties of $Δ^{(N)}$ as $N\to\infty$ in the subcritical regime $0<p<1$ where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of $Δ^{(N)}$ exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.
We observe that approximate copies of the function $Λ_{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} Λ_{n}(x)=\exp \left( -x_{1}-π\sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails of a large class of functions, with properties related to coordinate independence, convexity, homotheticity, and homogeneity. The function $Λ_{n}$ is an entropy maximizer (on a half-space) that is uniquely determined by a homogeneity condition together with rotational invariance about the $x_{1}$ direction and its behavior near the origin. These results are connected to the l…
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。
The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well a…
Tail representations for absolute, signed-power, inverse, and logarithmic moments
本文建立任意随机变量实阶矩的统一框架，通过累积分布函数和生存函数给出积分表示，涵盖连续、离散和混合分布，并应用于zeta和Skellam分布，同时得到对数矩表示，连接对数均值、拉普拉斯变换与Frullani恒等式。
We consider a random walk of $n$ steps starting at $x_0=0$ with a double exponential (Laplace) jump distribution. We compute exactly the distribution $p_{k,n}(Δ)$ of the gap $d_{k,n}$ between the $k^{\rm th}$ and $(k+1)^{\rm th}$ maxima in the limit of large $n$ and large $k$, with $α=k/n$ fixed. We show that the typical fluctuations of the gaps, which are of order $O( n^{-1/2})$, are described by a universal $α$-dependent distribution, which we compute explicitly. Interestingly, this distribution has an inverse cubic tail, which implies a non-trivial $n$-dependence of the moments of the gaps.…
We investigate the behavior near zero of the integrated density of states $\ell$ for random Schrödinger operators $Φ(-Δ) + V^ω$ in $L^2(\mathbb R^d)$, $d \geq 1$, where $Φ$ is a complete Bernstein function such that for some $α\in (0,2]$, one has $ Φ(λ) \asymp λ^{α/2}$, $λ\searrow 0$, and $V^ω(x) = \sum_{ \mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(ω) W(x-\mathbf{i})$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$ -C \leq\liminf_{λ\searrow 0} \frac{λ^{d/α}}{|\log …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。