Uniform large and moderate deviations for functional empirical processes

Abstract

For { X i } i ≥ 1 a sequence of i.i.d. random variables taking values in a Polish space Σ with distribution μ, we obtain large and moderate deviation principles for the processes { n −1 Σ [ nt] i = 1 δ X i ; t ≥ 0} n ≥ 1 and {n −1 2 Σ [nt] i = 1 (δ X i − μ); t ≥ 0} n ≥ 1 , respectively. Given a class of bounded functions F on Σ, we then consider the above processes as taking values in the Banach space of bounded functionals over F and obtain the corresponding (uniform over F), large and moderate deviation principles. Among the corollaries considered are functional laws of the iterated logarithm.