Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1154055 | Statistics & Probability Letters | 2007 | 5 Pages |
Abstract
Let k and m be two fixed positive integers with k⩾3 and 1ξ}. Define the hierarchical sequence of random variables Xn,1,n⩾0 by Xn+1,j=f(Xn,1+(j-1)k,Xn,2+(j-1)k,â¦,Xn,k+(j-1)k) with {X0,j,j⩾1} being independent random variables identically distributed as X0. In this note it is shown that Xn,1âdG(x)=ξI[λ1,â)(x)+(1-ξ)I[λ2,â)(x) and liminfnââXn,1=λ1 a.s. and limsupnââXn,1=λ2 a.s. It follows that limnââXn,1=λ a.s. if and only if λ1=λ2=λ. This result generalizes and improves Propositions 4.3 and 4.4 of Li and Rogers [1999. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9, 1175-1201].
Related Topics
Physical Sciences and Engineering
Mathematics
Statistics and Probability
Authors
Deli Li, Yongcheng Qi,