Almost sure convergence of solutions to non-homogeneous stochastic difference equation

We consider a non-homogeneous non-linear stochastic difference equationand its linear counterpartboth with initial value , non-random decaying free coefficient S n and independent random variables . We establish results on a.s. convergence of solutions X n to zero. Obtained necessary conditions tie together certain moments of the noise 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. Several examples are given to illustrate the ideas of the paper.

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