| | |
Summary: Large deviations of empirical processes
Miguel A. Arcones
Abstract. We give necessary and sufficient conditions for the large deviations
of empirical processes and of Banach space valued random vectors. We also
consider the large deviations of partial sums processes. The main tool used is
an isoperimetric inequality for empirical processes due to Talagrand.
April 13, 2004
1. Introduction
We study the (LDP) large deviation principle for different types of sequences of
empirical processes {Un(t) : t T}, where T is an index set. General references on
large deviations are Bahadur (1971), Varadhan (1984) and Deuschel and Stroock
(1989). 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}
n=1 such that n 0, and a function I : l(T) [0, ],
we say that {Un(t) : t T} satisfies the LDP with speed -1
n and with good rate
function I if:
|
