The random integral representation hypothesis revisited : new classes of s-selfdecomposable laws

Selfdecomposability and selfsimilarity: a concise primer

We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.

基于相关性重新审视高斯分布间的二次Wasserstein距离

Quadratic Wasserstein distance between Gaussian laws revisited with correlation

本文提出一种简单推导方法，基于正交矩阵使两高斯分布协方差矩阵共享同一相关矩阵，重新推导了两个高斯分布间二次Wasserstein距离的公式。

An Autoregressive Model with Semi-stable Marginals

The family of semi-stable laws is shown to be semi-selfdecomposable. Thus they qualify to model stationary first order autoregressive schemes. A connection between these autoregressive schemes with semi-stable marginals and semi-selfsimilar processes is given.

A Random Difference Equation with Dufresne Variables revisited

The Dufresne laws (laws of product of independent random variables with gamma and beta distributions) occur as stationary distribution of certain Markov chains $ X_n $ on $ R$ defined by: \begin{equation} X_n = A_n ( X_{n-1} + B_n ) \end{equation} where $ X_0 , (A_1,B_1),...,(A_n,B_n) $ are independent and the $(A_i,B_i)'$s are identically distributed. This paper generalizes an explicit example where $A$ is the product of two independent $β_{a,1} , β_{b,1} $ and $B \sim γ_1 $ or $ γ_2 $. Keywords: beta, gamma and Dufresne distributions,Markov chains, stationary distributions, hypergeometric di…

Selfdecomposability, perpetuity and stopping times

In the probability theory limit distributions (or probability measures) are often characterized by some convolution equations (factorization properties) rather than by Fourier transforms (the characteristic functionals). In fact, usually the later follows the first one. Equations, in question, involve the multiplication by the positive scalars $c$ or an action of the corresponding dilation $T_{c}$ on measures. In such a setting, it seems that there is no way for stopping times (or in general, for the stochastic analysis) to come into the "picture". However, if one accepts the view that the pri…

An introduction to a general records theory both for dependent data and high dimensions: revisited 2

The probabilistic investigation on record values and record times of a sequence of random variables defined on the same probability space has received much attention from 1952 to now. A great deal of such theory focused on \textit{iid} or independent real-valued random variables. There exists a few results for real-valued dependent random variables. Some papers deal also with multivariate random variables. But a large theory regarding vectors and dependent data has yet to be done. In preparation of that, the probability laws of records are investigated here, without any assumption on the depen…

关于用谱方法研究具有自可分极限定律的定量广义中心极限定理的一些注记

Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods

本文利用 Stein 方法、闭式技术和加权 Poincaré 型不等式，获得了自可分和对称 α-稳定概率测度的新稳定性估计，并计算了广义中心极限定理在 Wasserstein 型距离下的收敛速度，特别是 Cauchy 情形下的 $n^{-1}$ 收敛速度。