We consider a perturbed linear stochastic difference equationwith real coefficients a(n), g(n), σ (n), and independent identically distributed random variables ξ (n) having zero mean and unit variance. The sequence (a(n)) n∈N is K-periodic, where K is some positive integer, lim n→∞ g(n) =ĝ < ∞ and lim n→∞ σ (n)ξ (n + 1) = 0, almost surely. We establish conditions providing almost sure asymptotic periodicity of the solution X(n) for |L| = 1 and |L| < 1, where L :=
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.
MSC: 34F05; 39A30; 93E15; 37H10