A Local Limit Theorem for Robbins-Monro Procedure

TL;DR

This paper establishes a local limit theorem for the Robbins-Monro algorithm, providing detailed density convergence analysis using advanced probabilistic techniques, which enhances understanding of its asymptotic behavior.

Contribution

It introduces a local limit theorem for Robbins-Monro, extending Gaussian convergence results to density-level analysis with a novel parametrix approach for unbounded drifts.

Findings

Proves a local limit theorem for Robbins-Monro densities

Uses a parametrix technique for Markov chains with unbounded drifts

Provides detailed asymptotic density convergence results

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, a Gaussian convergence can be established for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.