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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 ofzero. We prove that general metric LDPs are preserved under uniformlycontinuous mappings. This allows us to transform the result of Borovkov andMogulskii into standard LDPs. We also give an explicit integral representationof the rate function they found. As an application, we extend the classicalCram'er theorem by proving an LPD for kernel-weighted sums of i.i.d. randomvectors in $R^d$.

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