EMPIRICAL PROCESSES, VAPNIK-CHERVONENKIS CLASSES AND
POISSON PROCESSES
Mark Durst
Richard M. Dudley
Abstract: For background of this paper see [2]. Given a probability space 
, let 
be the Gaussian process with mean 
, indexed by 
, and such that
(1)
Let
and suppose that, for all probability measures, (laws)
on
,
has a
version with bounded sample functions on
(For example, suppose
is a ”universal
Donsker class”.) Then, for some:
no set
of
elements has all its subsets
of the form
i.e.
is a Vapnik-Chervonenkis class. An example
shows that limit theorems for empirical measures need not hold uniformly over
a Vapnik-Chervonenkis class of measurable sets, unless further measurability is
assumed.
(2) For a law
on
the collection
of all subsets is a Donsker class if
and only if
(3)
For any probability space
suppose
is a P-Donsker class,
.
Let
be a Poisson point process with intensity measure
Then, as
converges in law, with respect to uniform convergence on
, to
the Gaussian process
with mean
and
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -
