##
**Exponential ergodicity of the bouncy particle sampler.**
*(English)*
Zbl 1467.60057

Summary: Nonreversible 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 [E. A. J. F. Peters and G. de With, “Rejection-free Monte Carlo sampling for general
potentials”, Phys. Rev. E 85, No. 2, Article ID 026703, 5 p. (2012; doi:10.1103/PhysRevE.85.026703)] is geometrically ergodic and we provide a central limit theorem for the associated ergodic averages. This holds essentially whenever the target satisfies a curvature condition and the growth of the negative logarithm of the target is at least linear and at most quadratic. For target distributions with thinner tails, we propose an original modification of this scheme that is geometrically ergodic. For targets with thicker tails, we extend the idea pioneered in [L. T. Johnson and C. J. Geyer, Ann. Stat. 40, No. 6, 3050–3076 (2012; Zbl 1297.60052)] in a random walk Metropolis context. We establish geometric ergodicity of the Bouncy Particle Sampler with respect to an appropriate transformation of the target. Mapping the resulting process back to the original parameterization, we obtain a geometrically ergodic piecewise deterministic Markov process.

### MSC:

60J25 | Continuous-time Markov processes on general state spaces |

65C05 | Monte Carlo methods |

65C40 | Numerical analysis or methods applied to Markov chains |

60F05 | Central limit and other weak theorems |