# Large deviations for distributions of sums of random variables: Markov chain method

Large deviations for distributions of sums of random variables: Markov chain method Let {ξ k } k=0 ∞ 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 $$P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\}$$ , 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 , p > 0, and exponential random variables with g(x) = x for x ≥ 0. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

# Large deviations for distributions of sums of random variables: Markov chain method

, Volume 46 (2) – Jul 23, 2010
24 pages

/lp/springer_journal/large-deviations-for-distributions-of-sums-of-random-variables-markov-GApyparKNz
Publisher
Springer Journals
Subject
Engineering; Systems Theory, Control; Information Storage and Retrieval; Electrical Engineering; Communications Engineering, Networks
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946010020055
Publisher site
See Article on Publisher Site

### Abstract

Let {ξ k } k=0 ∞ 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 $$P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\}$$ , 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 , p > 0, and exponential random variables with g(x) = x for x ≥ 0.

### Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 23, 2010

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