Large deviations for quadratic functionals of Gaussian
processes
Journal of Theoretical Probability 10 (1997) pp. 307-332
Abstract.
The Large Deviation Principle (LDP) is derived for several quadratic
additive
functionals of centered stationary Gaussian processes. For
example, the rate
function corresponding to
[ 1/T] ò0T Xt2 dt is the
Fenchel-Legendre transform
of
L(y)=-[ 1/(4p)]ò-¥¥ log(1-4py f(s))ds, where
Xt is a continuous time process with the bounded spectral
density f(s).
This spectral density condition is strictly weaker than
the one necessary for the LDP to hold for all
bounded continuous functionals.
Similar results are obtained for the energy of multivariate discrete-time
Gaussian processes and in the regime of moderate deviations, the latter
yielding the corresponding Central Limit Theorems.
Keywords.
large deviations, quadratic additive
functionals, Gaussian processes
AMS (1991) subject classification.
60F10
A. Dembo
Department of Mathematics
and Department of Statistics
Stanford University
Stanford, CA 94 305
E-Mail: amir@playfair.stanford.edu
W. Bryc
Department of Mathematics
University of Cincinnati
PO Box 210025
Cincinnati, OH 45221-0025
E-Mail: brycw@math.uc.edu