Strong diffusion approximations for recursive stochastic algorithms

Lai and Robbins (1978) prove strong diffusion approximations for the Robbins-Monro stochastic approximation algorithm. We show that similar strong approximations hold for stochastic algorithms at the level of generality proposed in the monograph of Benveniste, Metivier and Priouret (1990), wherein algorithms with generally discontinous right-hand sides driven by conditionally Markovian data are considered. The relevance of our result is demonstrated on an estimation algorithm with a discontinuous right-hand side which is used in data communication. The technique of adaptive delta modulation used in digital communication is considered.