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A large deviation local limit theorem

Published online by Cambridge University Press:  04 October 2011

R. A. Doney
R. A. Doney
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL
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Extract

The following elegant one-sided large deviation result is given by S. V. Nagaev in [2].

Theorem 0. Suppose that {Sn,n ≤ 0} is a random walk whose increments Xi are independent copies of X, where(X) = 0 and

Pr{X > x} ̃ x−αL(x) as x→ + ∞,

and where 1 < α < ∞ and L is slowly varying at ∞. Then for any ε > 0 and uniformly in x ≥ εn

Pr{Sn > x} ̃ n Pr{X > x} as n→∞.

It is the purpose of this note to point out that for lattice-valued random walks there is an analogous local limit theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 575 - 577
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Doney, R. A.. On last exit times for random walks. Stochastic Process. Appl. (to appear).Google Scholar
[2] Nagaev, S. V.. On the asymptotic behaviour of one-sided large deviation probabilities. Theory Probab. Appl. 26 (1982), 362–366.CrossRefGoogle Scholar