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arXiv · 2509.15410

Two-scale criteria for Poincar\'{e} and log-Sobolev inequalities with applications to Markov chain Monte Carlo

Vishwak Srinivasan
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Abstract

Given a collection of distributions $\{P_{y}\}$ and a mixing distribution $\rho$ supported over $\mathbb{R}^{d}$, we propose new sufficient conditions under which the mixture / joint distribution satisfies a Poincar\'{e} or log-Sobolev inequality. We develop these sufficient conditions in a unified manner using the framework of $\Phi$-Sobolev inequalities (Chafa\"{i}, 2004). The conditions that we develop in this work are satisfied by a variety of Markov chains, and consequently allows us to characterise the evolution of these functional inequalities for iterates generated by simulating these Markov chains. As a result, we obtain an clean error analysis for estimating a broad class of functionals using Markov chain Monte Carlo strategies along these Markov chains.

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Vishwak Srinivasan. 2025-09-18. Two-scale criteria for Poincar\'{e} and log-Sobolev inequalities with applications to Markov chain Monte Carlo. https://arxiv.org/abs/2509.15410

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