ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 1, pp. 56–69.
Pleiades Publishing, Inc., 2012.
Original Russian Text
M.V. Burnashev, Yu.A. Kutoyants, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 1, pp. 64–79.
On Large Deviations for Poisson Stochastic Integrals
M. V. Burnashev
and Yu. A. Kutoyants
Kharkevich Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow
Laboratoire Manceau de Math´ematiques, Universit´e du Maine, Le Mans, France
Received May 16, 2011; in ﬁnal form, October 25, 2011
Abstract—We obtain asymptotically exact estimates for large deviations of Poisson stochastic
integrals. We also ﬁnd a region where such an integral can be approximated by the correspond-
ing Gaussian random variable. In of all these results, we obtain nonasymptotic estimates for
1. INTRODUCTION AND MAIN RESULTS
We start with the following well-known result [1, Theorem 1]. Let X
identically distributed random variables with distribution function F (x). Let
,g(a)=− ln m(a)
(assuming that 0 <m(a) < ∞). If E X
< ∞ and a ≥ E X
≥ na) ≤ m
Moreover, if 0 <ε<m(a), then
(m(a) − ε)
= o (P(S
≥ na)) . (2)
The main contribution of Chernoﬀ [1, Theorem 1] is lower bound (2). Concerning upper
bound (1), Chernoﬀ notes in [1, Theorem 1] that this bound follows from the “extended Chebyshev
inequality” and refers to the book of Kolmogorov .
However, exactly estimate (1), which is very popular in information theory, was called there the
Chernoﬀ bound, though it would be more natural to call it the “exponential Chebyshev inequality.”
Relations (1) and (2) allow to get exact logarithmic asymptotics for the probability P(S
as n →∞. For any distribution F and any a ≥ E X,wehave
≥ na)=−g(a) − δ(F, a, n), (3)
where δ(F, a, n) ≥ 0andδ(F, a, n) → 0asn →∞(but not uniformly in F and a).
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.