摘要：We consider a perturbed linear stochastic difference equation 1 X ( n + 1 ) = a ( n ) X ( n ) + g ( n ) + σ ( n ) ξ ( n + 1 ) , n = 0 , 1 , … , X 0 ∈ R , $$ X(n+1)=a(n)X(n)+g(n)+\sigma(n)\xi(n+1), \quad n=0, 1, \dots, \qquad X_(\in\mathbb{R}, $$ with real coefficients a ( n ) $a(n)$ , g ( n ) $g(n)$ , σ ( n ) $\sigma(n)$ , and independent identically distributed random variables ξ ( n ) $\xi(n)$ having zero mean and unit variance. The sequence ( a ( n ) ) n ∈ N $(a(n) )_{n\in\mathbf {N}}$ is K-periodic, where K is some positive integer, lim n → ∞ g ( n ) = g ˆ < ∞ $\lim_{n\to\infty }g(n)=\hat{g}<\infty$ and lim n → ∞ σ ( n ) ξ ( n + 1 ) = 0 $\lim_{n\to\infty}\sigma(n) \xi(n+1)=0$ , almost surely. We establish conditions providing almost sure asymptotic periodicity of the solution X ( n ) $X(n)$ for L = 1 $ L =1$ and L < 1 $ L <1$ , where L : = ∏ i = 0 K − 1 a ( i ) $L:=\prod_{i=0}^{K-1}a(i)$ . A sharp result on the asymptotic periodicity of X ( n ) $X(n)$ is also proved. The results are illustrated by computer simulations.
关键词：stochastic difference equation ; asymptotically periodic solutions ; limiting behaviour of the solutions
