On almost sure asymptotic periodicity for scalar stochastic difference equations

被引:1
作者
Rodkina, Alexandra [1 ,2 ]
Rapoo, Eeva [2 ]
机构
[1] Univ West Indies, Dept Math, Mona Campus, Kingston 7, Jamaica
[2] Univ South Africa, Dept Stat, POB 392, ZA-0003 Pretoria, South Africa
关键词
stochastic difference equation; asymptotically periodic solutions; limiting behaviour of the solutions; STABILITY;
D O I
10.1186/s13662-017-1269-0
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We consider a perturbed linear stochastic difference equation X(n + 1) = a(n) X(n) + g(n) + sigma(n)xi (n + 1), n = 0,1,..., X-0 is an element of R, (1) with real coefficients a(n), g(n), sigma(n), and independent identically distributed random variables xi (n) having zero mean and unit variance. The sequence (a(n))(n is an element of N) is K-periodic, where K is some positive integer, lim(n -> 8) g(n) = (g) over bar < infinity and lim(n -> 8) sigma(n)xi(n + 1) = 0, almost surely. We establish conditions providing almost sure asymptotic periodicity of the solution X(n) for vertical bar L vertical bar = 1 and vertical bar L vertical bar < 1, where L := Pi(K-1)(i=0) a(i). A sharp result on the asymptotic periodicity of X(n) is also proved. The results are illustrated by computer simulations.
引用
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页数:29
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