ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 2, pp. 160–183.
Pleiades Publishing, Inc., 2010.
Original Russian Text
V.R. Fatalov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 2, pp. 66–90.
Large Deviations for Distributions of Sums
of Random Variables: Markov Chain Method
V. R. Fatalov
Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
Received July 1, 2008; in ﬁnal form, December 11, 2009
be a sequence of i.i.d. real-valued random variables, and let g(x)be
a continuous positive function. Under rather general conditions, we prove results on sharp
asymptotics of the probabilities P
, n →∞, and also of their conditional
versions. The results are obtained using a new method developed in the paper, namely, the
Laplace method for sojourn times of discrete-time Markov chains. We consider two examples:
standard Gaussian random variables with g(x)=|x|
, p>0, and exponential random variables
with g(x)=x for x ≥ 0.
1. INTRODUCTION AND FORMULATION OF THE MAIN RESULTS
Presently, large deviation theory for distributions of sums of independent random variables and
vectors is a vast area in limit theorems of probability theory (see [1, chs. 6–14; 2, ch. 8; 3; 4,
ch. 1; 5, Section 8.8]). In this area, rather many general and strong results are obtained, including
those for sums of Banach-valued random vectors (see [6–9] and also bibliography reviews in [6–8]).
Nevertheless, large deviation theory for distributions of sums of random vectors is still being rapidly
developed. This is due to a large number of diverse and highly complicated problems arising in
various areas of mathematics and applications which require development and application of large
deviation theory for sums. Here we mention numerous problems of ergodic theory, information
theory, mathematical statistics, and mathematical physics (see [10–14]).
Among existing methods of large deviation theory for sums, we distinguish the following: the
semiinvariant method [2, 3], saddle-point method [1,15,16], exponential Cram´er transform method
[4, 5], and Laplace method [6, 7, 17–21]. The ﬁrst three methods are designed for distributions
in ﬁnite-dimensional spaces R
, whereas the Laplace method is applicable to inﬁnite-dimensional
distributions too. For computations in particular problems, the Cram´er and Laplace methods are
rather convenient, where the main role is played by the so-called rate function, also referred to as
entropy function or deviation function.
The rate function, originating from Hamiltonian mechanics [22, p. 26], is often deﬁned as the
Legendre–Fenchel transform of some convex function. To large deviation theory, this function was
introduced in  (for k = 1),  (for k>1), and . Even for distributions in R, computing
this function is often a rather complicated problem.
In the present paper we propose a new general method for studying large deviation probabilities
for sums of i.i.d. random variables, namely, the Markov chain method. We show that in a number
Supported in part by the Russian Foundation for Basic Research, project no. 07-01-00077.