Fluctuations of bridges, reciprocal characteristics, and concentration of measure

TL;DR

This paper establishes new conditions based on reciprocal characteristics to control fluctuations of Markov process bridges, leading to non-asymptotic concentration inequalities and large deviation results for various stochastic processes.

Contribution

It introduces novel conditions on Markov generators that enable non-asymptotic analysis of bridge fluctuations, extending concentration inequalities and large deviations to new settings.

Findings

Concentration inequalities for gradient diffusion bridges.

Refined large deviation expansions for random walks on graphs.

New concentration results for pinned Poisson vectors.

Abstract

Conditions on the generator of a Markov process to control the fluctuations of its bridges are found. In particular, continuous time random walks on graphs and gradient diffusions are considered. Under these conditions, a concentration of measure inequality for the marginals of the bridge of a gradient diffusion and refined large deviation expansions for the tails of a random walk on a graph are derived. In contrast with the existing literature about bridges, all the estimates we obtain hold for non asymptotic time scales. New concentration of measure inequalities for pinned Poisson random vectors are also established. The quantities expressing our conditions are the so called \textit{reciprocal characteristics} associated with the Markov generator.