Limit Laws for Sums of Independent Random Products: the Lattice Case

作     者：Kabluchko, Zakhar 

作者机构：Univ Ulm Inst Stochast D-89069 Ulm Germany 

出 版 物：《JOURNAL OF THEORETICAL PROBABILITY》 (理论概率杂志)

年 卷 期：2012年第25卷第2期

页      面：424-437页

核心收录：

学科分类：07[理学] 0714[理学-统计学（可授理学、经济学学位）] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Random products Random exponentials Semi-stable laws Random energy model Triangular arrays Central limit theorem 

摘      要：Let {V-i, (j);(i, j) is an element of N-2} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products Zn = (Nn)Sigma(i=1) (n)Pi(j=1) e(Vi,j) as n, N-n - infinity have been investigated by a number of authors. Depending on the growth rate of N-n, the random variable Z(n) obeys a central limit theorem or has limiting alpha-stable distribution. The latter result is true for non-lattice V-i,V-j only. Our aim is to study the lattice case. We prove that although the (suitably normalized) sequence Z(n) fails to converge in distribution, it is relatively compact in the weak topology, and we describe its cluster set. This set is a topological circle consisting of semi-stable distributions.