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