线性递归分布方程

在各种应用概率背景下的一些问题———从算法的概率分析到统计物理,包括快速分类算法、自相似瀑布、无穷粒子系统和分支随机游动,常常引导我们研究线性递归分布方程Z=∑Ni=1AjZj的稳定分布解,其中“=”表示依分布相等,N和Zi是给定的实值随机变量,Zi之间相互独立且与{N,A1,A2,…}独立,Z和所有的Zi都是取值于R的未知的随机变量,且有共同的分布.对该方程的最基本的问题,如存在性、唯一性、非平凡解的渐进性质以及相关的光滑变换的迭代收敛性,给出了简要的概述.

On a linear recursive distributional equation

In certain problems in a variety of applied probability settings, from probability analysis of algorithms to statistical physics, including quicksort algorithm, self-similar cascades, infinite particles systems and branching random walks, we are often led to the study of stable-like laws, which satisfy a linear recursive distributional equation of the form Z=N∑i=1 AjZj in law, where N and Ti are given random real variables, Zi are independent of each other and independent of { N,A1 ,A2 ,... },and all the Z and Zi have the same law on R which is unknown. We give a short survey on the most fundamental problems about the equation, such as existence, uniqueness and asymptotic properties of nontrivial solutions, and convergence of iterations of the associated smoothing transformation.