On almost sure asymptotic periodicity for scalar stochastic difference equations

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.