Bridges of Markov counting processes: quantitative estimates

TL;DR

This paper analyzes the behavior of bridges in Markov counting processes, providing convexity characterizations, sharp distribution estimates, and convergence results as the process height increases.

Contribution

It introduces new criteria based on reciprocal characteristics to determine the shape and behavior of Markov counting process bridges.

Findings

Convexity of mean value linked to reciprocal bounds

Sharp estimates for marginal distributions

Convergence to deterministic curves at high levels

Abstract

In this paper we investigate the behavior of the bridges of a Markov counting process in several directions. We first characterize convexity(concavity) in time of the mean value in terms of lower (upper) bounds on the so called \textit{reciprocal characteristics}. This result gives a natural criterion to determine whether bridges are "lazy" or "hurried". Under the hypothesis of global bounds on the reciprocal characteristics we prove sharp estimates for the marginal distributions and a comparison theorem for the jump times. When the height of the bridge tends to infinity we show the convergence to a deterministic curve, after a proper rescaling.