 
Summary: Moderate deviations of empirical processes
Miguel A. Arcones
Abstract. We give necessary and sufficient conditions for the moderate devia
tions of empirical processes and of sums of i.i.d. random vectors with values in
a separable Banach space. Our approach is based in a characterization of the
large deviation principle using the large deviations of the finite dimensional
distributions plus an asymptotic exponential equicontinuity condition.
March 27, 2003
1. Introduction
We study the moderate deviations for different types of sequences of empirical
processes {Un(t) : t T}, where T is an index set. We also consider the moderate
deviations of sums of i.i.d. random vectors with values in a separable Banach space.
Our results are stated as functional large deviations with a Gaussian rate function.
General references on (functional) large deviations are Bahadur [4]; Varadhan
[23]; Deuschel and Stroock [9] and Shwartz and Weiss [21]. We consider stochastic
processes as elements of l(T), where T is an index set. l(T) is the Banach
space consisting of the bounded functions defined in T with the norm x =
suptT x(t). We will use the following definition:
Definition 1.1. Given a sequence of stochastic processes {Un(t) : t T}, a sequence
of positive numbers { n}
