In this paper, we study stochastic difference equation with asymptotic stability condition to the equilibrium. This equilibrium can be taken to be zero, without loss of generality. In this difference equation, linearization of the equation close to the equilibrium does not determine the asymptotic behaviour, because the terms which depend on the state of the system are o(x) as x → 0. The non-exponential convergence of solutions of stochastic difference and functional difference equations with bounded delay has been studied.