The problem of deriving frequency-domain conditions that ensure asymptotic stability with probability one (ASWP1) of discrete time non-linear feedback systems with state-dependent noise is considered. It is shown that if the noise is linearly related with the state of the system, a relatively simple frequency-domain inequality that guarantees ASWPl of the system, exists. The discrete stochastic version of the second method of Liapunov, along with Szegö-Kalman typo lemmae are used to derive such conditions, assuming that only sector information or both sector and slope information are available. The case of a difference system with white-noise parameter perturbation is also considered and an illustrative example is presented.