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Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

We consider a non-homogeneous nonlinear stochastic difference equation X_{n+1} = X_n (1 + f(X_n)ξ_{n+1}) + S_n, and its important special case X_{n+1} = X_n (1 + ξ_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables ξ_n. We establish results on \as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise ξ_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning the rate of decay of X_n to zero.