Franck Barthe, Patrick Cattiaux, Cyril Roberto
If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of $n$ independent copies, with good dependence in $n$.
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We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest stat…
We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.
Heavy Loads and Heavy Tails
本文利用变换方法研究重尾队列的平稳工作量，揭示了无限方差下尾部渐近性与重流量极限的联系，并证明减载等价性质可推广至重流量极限。
We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with (relatively) heavy tail of maximum distribution. The apparatus for tails comparison are embedding results for Orlicz and Grand Lebesgue Spaces (GLS).
In this work we present concentration inequalities for the sum $S_n$ of independent integer-valued not necessary indentically distributed random variables, where each variable has tail function that can be bounded by some power function with exponent $-α$. We show that when $0<α\leq 1$, then the sum does not have finite expectation, however, with high probability we have that $|S_n|=O\left(n^{1/α}\right)$. When $α>1$, then the sum $S_n$ is concentrated around its mean. Since the r.vs. that constitute the sum has tails, which can be bounded by some power function, it follows that results of thi…
In this paper the question of the extent to which truncated heavy tailed random vectors, taking values in a Banach space, retain the characteristic features of heavy tailed random vectors, is answered from the point of view of the central limit theorem.
Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum
本文研究了重尾随机变量的最小值为何能成为轻尾分布的条件，证明了只有特定的重尾分布才能生成轻尾的最小值，并扩展了相关结论。
The purpose of this paper is to show that the use of heavy-tailed distributions in Financial problems is theoretically baseless and can lead to significant misunderstandings. The reason for this the authors see in an incorrect interpretation of the concept of the distributional tail. In accordance with this, in applications it is necessary to use instead of the distributional tail the "smearing" of its central part. keywords: heavy tailed distributions, exponential tails, pre-limit theorems, Financial indexes
We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset.
This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with re…
We obtain an uniform tail estimates for natural normed sums of independent random variables (r.v.) with regular varying tails of distributions. We give also many examples on order to show the exactness of offered estimates and discuss some applications in the method Monte-Carlo and statistics, and obtain the sufficient conditions for Central and stable limit theorem in the Banach space of continuous function. There are considered a slight generalization on a random variables with super-heavy tails and martingale difference scheme.
We consider a Wigner matrix $A$ with entries tail decaying as $x^{-α}$ with $2<α<4$ for large $x$ and study fluctuations of linear statistics $N^{-1}\operatorname{Tr}φ(A)$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $α< 2$) and light-tailed matrices (i.e. $α> 4$). This paper fills in the gap of understanding for $2<α<4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-…
Heavy-tailed random vectros: theory and applications
本文研究了几类多变量厚尾分布及其闭包性质与应用，探讨了正递减分布类与其他分布类的交集。
Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.
As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, $\E e^{λZ_1}=\infty$ for every $λ>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur.
Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $ρ$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a …
This article studies asymptotic approximations of ruin probabilities of multivariate random walks with heavy-tailed increments. Under our assumptions, the distributions of the increments are closely connected to multivariate subexponentiality and admit dependence between components. Keywords: subexponential distribution, ruin probability, multivariate random walk
We investigate maxima of linear processes with i.i.d. heavy-tailed innovations and random coefficients. Using the point process approach we derive functional convergence of the partial maxima stochastic process in the space of non-decreasing càdlàg functions on $[0,1]$ with the Skorohod $M_{1}$ topology.
It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviation scale which we call hidden large deviation. We apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of severe congestion times for (buffered) queues. We conclude with simulated examples to verify our results.
We investigate a way of comparing and classifying tails of random variables. Our approach extends the notion of classical indices, such as exponential and moment indices, which are widely used measuring heaviness of tail functions. A non-parametric risk measure applicable for all heavy-tailed random variables is obtained as a concave function that represents the decay speed of tail function. Many key properties of the distribution of a random variable are encoded into this function, which enables a new way to estimate tails. The latter half of the paper is devoted to numerous examples illustra…
Any random variable with right-unbounded distributional support is the minimum of independent and very heavy-tailed random variables
本文证明任意具有右无界支撑的轻尾随机变量均可表示为两个独立重尾随机变量的最小值，且该结论可推广至有限个独立变量及相依情形。
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…
Large Deviations for SDE driven by Heavy-tailed Lévy Processes
本文研究了一类由有界漂移和重尾Lévy过程驱动的一维随机微分方程，在无指数可积性条件下证明了其满足具有离散速率函数和对数速度的弱大偏差原理，但不满足完整的大偏差原理。
This work deals with the divisible sandpile model when an initial configuration sampled from a heavy-tailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and non-stabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an $α$-stable random distribution.
Let $\{X_1, X_2, ... \}$ be a sequence of dependent heavy-tailed random variables with distributions $F_1, F_2,...$ on $(-\infty,\infty)$, and let $τ$ be a nonnegative integer-valued random variable independent of the sequence $\{X_k, k \ge 1\}$. In this framework, we study the asymptotic behavior of the tail probabilities of the quantities $X_{(n)} = \max_{1\le k \le n} X_k$, $S_n =\sum_{k=1}^n X_k$ and $S_{(n)}=\max_{1\le k\le n} S_k$ for $n>1$, and for those of their randomized versions $X_{(τ)}$, $S_τ$ and $S_{(τ)}$. We also consider applications of the results obtained to some commonly-us…
Computation of extreme quantiles and tail-based risk measures using standard Monte Carlo simulation can be inefficient. A method to speed up computations is provided by importance sampling. We show that importance sampling algorithms, designed for efficient tail probability estimation, can significantly improve Monte Carlo estimators of tail-based risk measures. In the heavy-tailed setting, when the random variable of interest has a regularly varying distribution, we provide sufficient conditions for the asymptotic relative error of importance sampling estimators of risk measures, such as Valu…
We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.
Asymptotic behavior of the extremal position in a multi-type branching random walk with heavy-tailed displacements
本文研究了位移具有重尾的多类型分支随机游动，探讨了不可约与可约情形下最右粒子的渐近行为，并为不同设定确定了正确的归一化。
Some notes on moment inequalities for heavy-tailed distributions
本文研究重尾（尤其 Pareto 型）分布的矩与尾部关系，在正则性条件下验证结论锐性，并给出任意阶 $d$ 多项式混沌的集中界。
We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology.
Sub-Gaussian estimates for the natural random walk is typical of many regular fractal graphs. Subordination shows that there exist heavy tailed jump processes whose jump indices are greater than or equal to two. However, the existing machinery used to prove heat kernel bounds for such heavy tailed random walks fail in this case. In this work we extend Davies' perturbation method to obtain transition probability bounds for these anomalous heavy tailed random walks. We prove global upper and lower bounds on the transition probability density that are sharp up to constants. An important feature o…
This paper deals with symmetric random matrices whose upper diagonal entries are obtained from a linear random field with heavy tailed noise. It is shown that the maximum eigenvalue and the spectral radius of such a random matrix with dependent entries converge to the Frechét distribution after appropriate scaling. This extends a seminal result of Soshnikov(2004) when the tail index is strictly less than one.
Concentration inequalities for subgraph counts in random geometric graphs built over Poisson point processes are proved. The estimates give upper bounds for the probabilities $\mathbb{P}(N\geq M +r)$ and $\mathbb{P}(N\leq M - r)$ where $M$ is either a median or the expectation of a subgraph count $N$. The bounds for the lower tail have a fast Gaussian decay and the bounds for the upper tail satisfy an optimality condition. A special feature of the presented inequalities is that the underlying Poisson process does not need to have finite intensity measure. The tail estimates for subgraph counts…
Tail Asymptotics of Cluster Sizes in Multivariate Heavy-Tailed Hawkes Processes
研究多元重尾Hawkes过程簇大小的尾渐近性，通过分析多类型分支过程的分布不动点方程，揭示多元隐正则变化特征及其由大跳跃决定的机制。
Results have been moved to a published article, see arXiv:0812.2669v4[math.PR]
For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.
Quantitative bounds for large deviations of heavy tailed random variables
本文研究了独立同分布重尾随机变量之和的大偏差渐近近似，量化了该近似的误差，并刻画了在和值极大条件下单个和项的分布律。
We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.
We consider a two-node fluid network with batch arrivals of random size having a heavy-tailed distribution. We are interested in the tail asymptotics for the stationary distribution of a two-dimensional queue-length process. The tail asymptotics have been well studied for two-dimensional reflecting processes where jumps have either a bounded or an unbounded light-tailed distribution. However, presence of heavy tails totally changes the asymptotics. Here we focus on the case of strong stability where both nodes release fluid with sufficiently high speeds to minimise their mutual influence. We s…
A large deviations principle is established for the joint law of the empirical measure and the flow measure of a renewal Markov process on a finite graph. We do not assume any bound on the arrival times, allowing heavy tailed distributions. In particular, the rate functional is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behavior highly different from what one may guess with a heuristic Donsker-Varadhan analysis of the problem.
The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter $α\in(0,1]$. We turn to the case of very heavy tailed reproduction distribution $α=0$ assuming Zubkov's regularity condition with parameter $β\in(0,\infty)$. Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.
In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time $W$ in the $GI/GI/2$ FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability ${\bf P}\{W>x\}$. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load.
Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distribu…
Importance sampling algorithms for heavy-tailed random walks are considered. Using a specification with algorithms based on mixtures of the original distribution with some other distribution, sufficient conditions for obtaining bounded relative error are presented. It is proved that mixture algorithms of this kind can achieve asymptotically optimal relative error. Some examples of mixture algorithms are presented, including mixture algorithms using a scaling of the original distribution, and the bounds of the relative errors are calculated. The algorithms are evaluated numerically in a simple …
In this paper we consider a stochastic model of perpetuity-type. In contrast to the classical affine perpetuity model of Kesten [12] and Goldie [8] all discount factors in the model are mutually independent. We prove that the tails of the distribution of this model are regularly varying both in the univariate and multivariate cases. Due to the additional randomness in the model the tails are not pure power laws as in the Kesten-Goldie setting but involve a logarithmic term.
We present novel martingale concentration inequalities for martingale differences with finite Orlicz-$ψ_α$ norms. Such martingale differences with weak exponential-type tails scatters in many statistical applications and can be heavier than sub-exponential distributions. In the case of one dimension, we prove in general that for a sequence of scalar-valued supermartingale difference, the tail bound depends solely on the sum of squared Orlicz-$ψ_α$ norms instead of the maximal Orlicz-$ψ_α$ norm, generalizing the results of Lesigne & Volný (2001) and Fan et al. (2012). In the multidimensional ca…
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.
Concentration Inequalities for Sub-Weibull Random Tensors
本文提出了一种针对子韦布分布随机张量的集中不等式理论，通过广义最大不等式和鞅分析，建立了欧几里得函数的集中界限，并揭示了亚高斯与重尾区域之间的相变现象。
Distributions with a heavy tail are difficult to estimate. If the design of a scheduling policy is sensitive to the details of heavy tail distributions of the service times, an approximately optimal solution is difficult to obtain. This paper shows that the optimal scheduling of an M/G/1 queue with heavy tailed service times does not present this difficulty and that an approximately optimal strategy can be derived by truncating the distributions.
DynaSPLIT: Strong tail-optimal scheduling for the M/G/n with heavy-tailed job sizes
本文针对作业规模为重尾分布的M/G/n队列，提出首个适用于高负载的强尾最优调度策略DynaSPLIT，填补了该场景高负载下强尾最优调度的研究空白，证明该队列全稳定区域均可实现强尾最优性。
We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy $\mathbb{P}(X>t) \leq {\rm e}^{- I(t)}$, where $I: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow α\in [0, \infty)$ as $t \rightarrow \infty$. Our main theorem can not only recover some of the existing results, such as the concentration of the sum of subWeibull random variables, but it can also produce new results for th…
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…
The choice of the increment distribution is crucial for the random-walk Metropolis-Hastings (RWM) algorithm. In this paper we study the optimal choice in high-dimension setting among all possible increment distributions. The conclusion is rather counter intuitive, but the optimal rate of convergence is attained by the usual choice, the normal distribution as the increment distribution. In particular, no heavy-tailed increment distribution can improve the rate.
We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavy-tailed random conductance Laplacian to the first $k$ eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike theorem to show that the $k$th eigenvector is close to the principal eigenvector of an auxiliary spectral problem.
We are trying to give a mathematically correct definition of outliers. Our approach is based on the distance between two last order statistics and appears to be connected to the law of large numbers. Key words: outliers, law of large numbers, heavy tails, stability index.
Tail Asymptotic of Heavy-Tail Risks with Elliptical Copula
本文研究具有重尾边缘和 I 型椭圆相依结构的多元分布，推导了渐近尾部风险概率并刻画多元正则变化性质，揭示了衰减速率与尾部集合及椭圆 Copula 协方差矩阵的关系，并通过数值模拟和真实保险损失数据验证了理论结果。
Concentration and moment inequalities for sums of independent heavy-tailed random matrices
本文针对仅存在低阶有限矩的独立重尾随机矩阵和，证明了依赖有效秩等内在维度的Fuk-Nagaev型与Rosenthal型不等式，并通过多个应用验证了结果优势。
Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e., the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.
We consider the problem of estimation of the drift parameter of an ergodic Ornstein--Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval $[0,T]$, $T\to\infty$. We prove that the statistical model is locally asymptotic mixed normal and the maximum likelihood estimator is asymptotically efficient.
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