Local Limit Theorems and Strong Approximations for Robbins-Monro Procedures

TL;DR

This paper develops local limit theorems and strong approximation results for Robbins-Monro procedures, enhancing understanding of their stochastic behavior through advanced probabilistic techniques.

Contribution

It introduces local limit theorems for Robbins-Monro algorithms using parametrix methods, addressing challenges posed by unbounded drifts.

Findings

Established local limit theorems for Robbins-Monro procedures.

Provided strong approximation results for the stochastic process.

Addressed technical difficulties due to unbounded drifts.

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.