Large Deviations for Processes on Half-Line: Random Walk and Compound Poisson

TL;DR

This paper proves an extended large deviation principle for random walk and compound Poisson processes on a half-line, using a modified Borovkov metric, under certain exponential moment conditions, providing more precise probabilistic estimates.

Contribution

It establishes a more precise large deviation principle in a specialized function space with a modified metric, extending previous results under exponential moment conditions.

Findings

LDP holds in the space of functions of finite variation with the modified Borovkov metric.

The LDP is more precise than the uniform convergence metric version.

Results apply to random walk and compound Poisson processes under Cramér's condition.

Abstract

We establish, under the Cramer exponential moment condition in a neighbourhood of zero, the Extended Large Deviation Principle for the Random Walk and the Compound Poisson processes in the metric space $\V$ of functions of finite variation on $[0,\infty)$ with the modified Borovkov metric $\r(f,g)= \r_\B(\hat{f},\hat{g}) $, where $ \hat f(t)= f(t)/(1+t)$, $t\in \R$, and $\r_\B$ is the Borovkov metric. LDP in this space is "more precise" than that with the usual metric of uniform convergence on compacts.