# Exponential Ergodicity of the Bouncy Particle Sampler

## TL;DR

This paper establishes verifiable conditions for the geometric ergodicity of the Bouncy Particle Sampler, extending its applicability to a wider class of target distributions including those with various tail behaviors.

## Contribution

The paper provides new verifiable conditions for geometric ergodicity of the Bouncy Particle Sampler and introduces modifications for targets with different tail behaviors.

## Key findings

- Conditions for geometric ergodicity under certain tail decay rates.
- A modified scheme for thin-tailed distributions ensuring ergodicity.
- Transformation approach for thick-tailed distributions like t-distributions.

## Abstract

Non-reversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in \cite{P_dW_12} is geometrically ergodic. This holds whenever the target satisfies a curvature condition and has tails decaying at least as fast as an exponential and at most as fast as a Gaussian distribution. This allows us to provide a central limit theorem for the associated ergodic averages. When the target has tails thinner than a Gaussian distribution, we propose an original modification of this scheme that is geometrically ergodic. For thick-tailed target distributions, such as $t$-distributions, we extend the idea pioneered in \cite{J_G_12} in a random walk Metropolis context. We apply a change of variable to obtain a transformed target satisfying the tail conditions for geometric ergodicity. By sampling the transformed target using the Bouncy Particle Sampler and mapping back the Markov process to the original parameterization, we obtain a geometrically ergodic algorithm.