A remark on normalizations in a local principle of large deviations

TL;DR

This paper investigates large deviations for a continuous-time birth-and-death process with power-law transition rates, exploring normalization schemes that preserve the large deviation functional's integral form.

Contribution

It introduces various normalization schemes ensuring the large deviation functional maintains its natural integral form for such processes.

Findings

Established exponential asymptotics for the probability of normalized paths near a given function.

Proposed normalization schemes that preserve the large deviation functional's integral form.

Abstract

This work is a continuation of [7]. We consider a continuous-time birth-and-death process in which the transition rates have an asymptotical power-law dependence upon the position of the process. We establish rough exponential asymptotic for the probability that a sample path of a normalized process lies in a neighborhood of a given nonnegative continuous function. We propose a variety of normalization schemes for which the large deviation functional preserves its natural integral form.