# Hypotheses testing and posterior concentration rates for semi-Markov processes

## TL;DR

This paper develops a Bayesian framework to analyze the asymptotic behavior of semi-Markov processes, providing conditions for posterior concentration rates and constructing robust statistical tests.

## Contribution

It introduces nonparametric Bayesian methods for semi-Markov processes and derives posterior concentration rates, extending existing theory to continuous time and general state spaces.

## Key findings

- Established posterior concentration rates for semi-Markov kernels
- Constructed robust tests between Hellinger balls around kernels
- Applied methods to discrete-time and finite state space cases

## Abstract

In this paper, we adopt a nonparametric Bayesian approach and investigate the asymptotic behavior of the posterior distribution in continuous time and general state space semi-Markov processes. In particular, we obtain posterior concentration rates for semi-Markov kernels. For the purposes of this study, we construct robust statistical tests between Hellinger balls around semi-Markov kernels and present some specifications to particular cases, including discrete-time semi-Markov processes and finite state space Markov processes. The objective of this paper is to provide sufficient conditions on priors and semi-Markov kernels that enable us to establish posterior concentration rates.