Uniform ergodicity and spectral gap of importance-tempered MCMC with heavy-tailed targets

Abstract

We study the importance-tempered random-walk Metropolis (RWM) algorithm on ${\mathbb{R}}^d$, which targets the tempered density ${\pi }_{\beta }(x)\propto \pi (x{)}^{\beta }$ with $\beta \in (0,1)$ and corrects for the resulting bias by importance weighting. The dynamics of this sampler can be described by a continuous-time Markov chain whose embedded chain is the tempered RWM. We prove that when the tails of $\pi$ decay at rate $\Vert x{\Vert }^{-\gamma }$, this continuous-time chain is uniformly ergodic if and only if $d/\gamma <\beta <(\gamma -2)/\gamma$. Further, the global spectral gap of this chain can be controlled via the spectral gap of the chain restricted to a ball with radius growing polynomially in $d$. Our theory and simulation studies show that, with a suitable choice of $\beta$, importance tempering can substantially reduce sensitivity to initialization and significantly improve efficiency over the standard RWM.

arXiv commentThis manuscript is a minor revision of an earlier version submitted to the journal on April 2, 2026