On Cubic Difference Equations with Variable Coefficients and Fading Stochastic Perturbations

Abstract

We consider the stochastically perturbed cubic difference equation with variable coefficientsHere (ξ n ) n∈N is a sequence of independent random variables, and (ρ n ) n∈N and (h n ) n∈N are sequences of nonnegative real numbers. We can stop the sequence (h n ) n∈N after some random time N so it becomes a constant sequence, where the common value is an F N -measurable random variable. We derive conditions on the sequences (h n ) n∈N , (ρ n ) n∈N and (ξ n ) n∈N , which guarantee that lim n→∞ x n exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x 0 ∈ R.