Abstract
A random discrete-time system { x n }, n = 0, 1, 2, … is called stochastically stable if for every ϵ > 0 there exists a λ > 0 such that the probability P[(sup n ∥ x n ∥) > ϵ] < ϵ whenever P[∥ x 0 ∥ > λ] < λ. A system is shown stochastically stable if some local Lyapunov function V(·) satisfies the supermartingale definition on { V( x n )} in a neighborhood of the origin; earlier proofs of stochastic stability require additional restrictions. A criterion for x n → 0 almost surely is developed. It consists of a global inequality on { U( x n )} stronger than the supermartingale defining inequality, but applied to a U(·) that need not be a Lyapunov function. The existence of such a U(·) is exhibited for a stochastically unstable nontrivial stochastic system. This indicates that our criterion for x n → 0 is “tight,” and that the two stability concepts studied are substantially distinct.
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