Local Limit Theorems and Strong Approximations for Robbins-Monro Procedures

TL;DR

This work develops a rigorous framework for strong approximations of Robbins-Monro stochastic approximation by linking RM transitions to a Gaussian diffusion via local limit theorems and the parametrix method. A carefully constructed truncated RM process is introduced to handle unbounded drift, and a diffusion limit is derived for the renormalized RM dynamics. The authors establish precise, beta-dependent rates bounding total variation and Hellinger distances between RM and diffusion transition densities, both on increasing time grids and in joint distributions, yielding strong approximation results with explicit convergence rates. The methodology combines parametrix expansions, Edgeworth-type density approximations, and truncation arguments to deliver a robust density-based comparison, with potential applications to stochastic optimization and other gradient-based procedures in high dimensions.

Abstract

The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, Gaussian limit distributions approximate the stochastic performance of the algorithm. Here, we are interested in strong approximations for Robbins-Monro procedures. The main tool for getting them are local limit theorems, that is, studying the convergence of the density of the algorithm. The analysis relies on a version of parametrix techniques for Markov chains converging to diffusions. The main difficulty that arises here is the fact that the drift is unbounded.

Theorems & Definitions (28)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Theorem 2.1
• Corollary 2.4
• Proposition 2.5
• Proposition 2.6
• Theorem 2.2
• Lemma 2.7
• Lemma 2.8