Stability of stochastic approximation under verifiable conditions

Abstract. In this paper we address the problem of the stability of the stochastic approximation procedure. The stability of such algorithms is known to rely heavily on the growth of the mean field at the boundary of the parameter set. The typical conditions are in practice either difficult to check or not satisfied, even for simple models. The most popular technique to get rid of this problem is to constrain the parameter to a compact subset in the parameter space. We propose and analyse an alternative, based on projection on adaptive truncation sets, extending previous works in this direction. The stability- with probability one- of the scheme is proved under a set of verifiable assumptions. We illustrate these claims in the so-called controlled Markovian setting and provide two substantial examples. The first example is related to the minimum prediction error estimation of the parameters of stable and invertible ARMA processes and the second example is related to adaptive Markov chain Monte Carlo algorithms. Key words. Stochastic approximation, state-dependent noise, randomly varying truncation, Adaptive Markov Chain Monte Carlo. AMS subject classifications. 62L20,90C15 1. Introduction. I