摘要
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.
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-llke laws, which satisfy a linear recursive distributional equation of the form Z = ∑i=1NAjZj in law, where Nand T~ 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.
出处
《长沙理工大学学报(自然科学版)》
CAS
2006年第3期91-97,共7页
Journal of Changsha University of Science and Technology:Natural Science
关键词
分布方程
函数方程
光滑变换
无穷粒子系统
分支过程
分支随机游动
乘积瀑布
快速分类算法
distributional equation
functional equation
smoothing transformation
infinite particle systems
branching processes
branching random walks
multiplicative cascades
algorithm quicksort