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arXiv · 2304.10673

Local Limit Theorems and Strong Approximations for Robbins-Monro Procedures

Valentin Konakov·Enno Mammen·Lorick Huang
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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.

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Valentin Konakov, Enno Mammen, Lorick Huang. 2023-04-20. Local Limit Theorems and Strong Approximations for Robbins-Monro Procedures. https://arxiv.org/abs/2304.10673

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