ISSN 0032-9460, Problems of Information Transmission, 2006, Vol. 42, No. 1, pp. 46–63.
Pleiades Publishing, Inc., 2006.
Original Russian Text
V.R. Fatalov, 2006, published in Problemy Peredachi Informatsii, 2006, Vol. 42, No. 1, pp. 52–71.
Exact Asymptotics of Large Deviations
of Stationary Ornstein–Uhlenbeck Processes
V. R. Fatalov
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Received May 25, 2005
Abstract—We prove a general result on the exact asymptotics of the probability
dt > u
as u →∞,wherep>0, for a stationary Ornstein–Uhlenbeck process η
(t), i.e., a Gaussian
Markov process with zero mean and with the covariance function E η
t, s ∈ R, γ>0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation
of constants is reduced to solving an extreme value problem for the rate function and studying
the spectrum of a second-order diﬀerential operator of the Sturm–Liouville type. For p =1
and p = 2, explicit formulas for the asymptotics are given.
1. INTRODUCTION AND FORMULATION OF MAIN RESULTS
Ornstein–Uhlenbeck processes ﬁnd numerous applications in various areas of mathematics,
physics, and engineering (see [1–8]). Though these processes are connected with a Wiener pro-
cess, ﬁnding exact distributions of various nonlinear functionals of these processes is nevertheless
a very complicated problem. Therefore, the problem of ﬁnding exact asymptotics of such distribu-
tions in the region of large or small values is quite natural. In particular, of both theoretical and
practical interest are asymptotics of distributions of such functionals as
(referred to as the L
(t), t ∈ R,beastationary Ornstein–Uhlenbeck process of order γ>0. Recall that this is
a Gaussian Markov process with a.s. continuous trajectories, zero mean, and covariance function
(t, s)=E η
,t,s∈ R; (1.1)
see [9, ch. 4, Section 8, Example 8.1], [10, ch. 3, Exercise 27], and [10, ch. 8, Section 14].
Below, when speaking about Ornstein–Uhlenbeck processes, we mean positive-order processes
In the case of integral functionals, the following three problems are of both theoretical and
Supported in part by the Russian Foundation for Basic Research, project no. 04-01-00700.