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.
It is well known that the Laplace-Stieltjes transform of a nonnegative random variable (or random vector) uniquely determines its distribution function. We extend this uniqueness theorem by using the Muntz-Szasz Theorem and the identity for the Laplace-Stieltjes and Laplace-Carson transforms of a distribution function. The latter appears for the first time to the best of our knowledge. In particular, if X and Y are two nonnegative random variables with joint distribution H, then H can be characterized by a suitable set of countably many values of its bivariate Laplace-Stieltjes transform. The …
We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are nondecreasing. We determine the Laplace-Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace-Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.
In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $α$ moments for $2<α< 4$. We obtain a closed form $α$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covarian…
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 this note, we study the ultimate ruin probabilities of a real-valued L{é}vy process X with light-tailed negative jumps. It is well-known that, for such L{é}vy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to infinity. Under the additional assumption that X has integrable positive jumps, we show how a finer analysis of the Laplace exponent gives in fact a complete description of the bounds on the probability of ruin for this class of L{é}vy processes. This leads to the identification …
We study the joint limit behavior of sums, maxima and $\ell^p$-type moduli for samples taken from an $\mathbb{R}^d$-valued regularly varying stationary sequence with infinite variance. As a consequence, we can determine the distributional limits for ratios of sums and maxima, studentized sums, and other self-normalized quantities in terms of hybrid characteristic functions and Laplace transforms. These transforms enable one to calculate moments of the limits and to characterize the differences between the iid and stationary cases in terms of indices which describe effects of extremal clusterin…
We analyze fluctuations of random walks with generally distributed increments. Integral representations for key performance measures are obtained by extending an inversion theorem of Hewitt [11] for Laplace-Stieltjes transforms. Another important part of the anal- ysis involves the so-called harmonic measures associated to the distribution of the increment of the walk. It is also pointed out that such representations can be explicitly calculated, if one assumes a form of rational structure for the increment transform. Applications include, but are not restricted to, queueing and insurance risk…
Distributional equation is an important tool in the characterization theory because many characteristic properties of distributions can be transferred to such equations. Using a novel and natural approach, we retreat a remarkable distributional equation whose corresponding functional equation in terms of Laplace-Stieltjes transform is of the Poincare type. The necessary and sufficient conditions for the equation to have a unique distributional solution with finite variance are provided. This complements the previous results which involve at most the mean of the distributional solution. Besides…
The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat P. L. Hsu's (1951, J. Chinese Math. Soc., Vol. 1, pp. 257-280) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace-Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely …
We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equat…
We report some properties of heavy-tailed Sibuya-like distributions related to thinning, self-decomposability and branching processes. Extension of the thinning operation of on-negative integer-valued random variables to scaling by arbitrary positive number leads to a new class of probability distributions with generating function $Q(w)$ expressible as a Laplace transform $φ(1-w)$ and probability mass function $p_n$ satisfying simple one step recurrence relation between $p_{n+1}$ and $p_n$. We show that the compound Poisson-Sibuya and the shifted Sibuya distributions belong to this class. Usin…
This work is devoted to a vast extension of Sanov's theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new …
A Constructive Approach to $q$-Gaussian Distributions: $α$-Divergence as Rate Function and Generalized de Moivre-Laplace Theorem
Graduate School of Informatics, Chiba University（千叶大学信息科学研究院）；Istituto dei Sistemi Complessi (ISC-CNR) c/o, Politecnico di Torino（都灵理工大学内的复杂系统研究所（ISC-CNR））
In this paper we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical r…
