Convergence rates of Gibbs measures with degenerate minimum

DOI10.3150/21-BEJ1424arXiv2101.11557WikidataQ114038742 ScholiaQ114038742MaRDI QIDQ6359206

Pierre Bras

Publication date: 27 January 2021

Abstract: We study convergence rates for Gibbs measures, with density proportional to $e^{-f(x)/t}$, as $tightarrow0$ where $f:mathbb{R}^{d}ightarrowmathbbR$ admits a unique global minimum at $x^{s}tar$. We focus on the case where the Hessian is not definite at $x^{s}tar$. We assume instead that the minimum is strictly polynomial and give a higher order nested expansion of $f$ at $x^{s}tar$, which depends on every coordinate. We give an algorithm yielding such a decomposition if the polynomial order of $x^{s}tar$ is no more than $8$, in connection with Hilbert's $17^{extth}$ problem. However, we prove that the case where the order is $10$ or higher is fundamentally different and that further assumptions are needed. We then give the rate of convergence of Gibbs measures using this expansion. Finally we adapt our results to the multiple well case.

Mathematics Subject Classification ID

Central limit and other weak theorems (60F05) Stochastic programming (90C15) Convergence of probability measures (60B10)

This page was built for publication: Convergence rates of Gibbs measures with degenerate minimum

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6359206)