Contraction principle for trajectories of random walks and Cramér’s theorem for kernel-weighted sums

Abstract

In 2013 A.A. Borovkov and A.A. Mogulskii proved a weaker-than-standard 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 $R^d$.