A large deviations principle for birth-death processes with a linear rate of downward jumps

TL;DR

This paper establishes a large deviation principle for birth-death processes with linearly increasing death rates and sub-linearly increasing birth rates, revealing how process parameters influence deviation probabilities.

Contribution

It introduces a large deviation principle for birth-death processes with specific growth conditions on birth and death rates, expanding theoretical understanding.

Findings

Large deviation functional depends on process parameters

Results hold under various scaling and normalization schemes

Provides insights into the asymptotic behavior of birth-death processes

Abstract

Birth-death processes form a natural class where ideas and results on large deviations can be tested. In this paper, we derive a large deviation principle under the assumption that the rate of a jump down (death) is growing asymptotically linearly with the population size, while the rate of a jump up (birth) is growing sub-linearly. We establish a large deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large deviation probabilities. The results show interesting features of dependence of the large deviation functional upon the parameters of the process and the forms of scaling and normalization.