CONTENTS

Abstract. In this paper, we establish the almost sure asymptotic stability and decay results for solutions of a autonomous scalar difference equation with a non–hyperbolic equilibrium at the origin, which is perturbed by a random term with a fading state–independent intensity. In particular, we show that when the unbounded noise has tails which fade more quickly than polynomially, the state–independent perturbation dies away at a sufficiently fast polynomial rate in time, and when the autonomous difference equation has a polynomial nonlinearity at the origin, then the almost sure polynomial rate of decay o