Contraction principle for trajectories of random walks and Cramer's theorem for kernel-weighted sums

TL;DR

This paper extends large deviations principles for random walk trajectories to standard forms and applies these results to establish a Cramér-type theorem for kernel-weighted sums of i.i.d. vectors.

Contribution

It shows that metric LDPs are preserved under uniformly continuous mappings, transforming weaker results into standard LDPs, and extends Cramér's theorem to kernel-weighted sums.

Findings

Metric LDPs are preserved under uniformly continuous maps.

Explicit integral representation of the rate function.

Established LDP for kernel-weighted sums of i.i.d. vectors.

Abstract

In 2013 A.A. Borovkov and A.A. Mogulskii proved a weaker-than-standard "metric" large deviations principle (LDP) for trajectories of random walks in $R^d$ whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cram'er theorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in $R^d$.