Uniform large and moderate deviations for functional empirical processes

For {Xi}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 δXi; t ≥ 0}n ≥ 1 and {n−12 Σ[nt]i = 1 (δXi − 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.