## The Annals of Probability

### Approximate Tail Probabilities for the Maxima of Some Random Fields

David Siegmund

#### Abstract

For random walks $\{S_n\}$ whose distribution can be embedded in an exponential family, large-deviation approximations are obtained for the probability that $\max_{0\leq i < j\leq m}(S_j - S_i) \geq b$ (i) conditionally given $S_m$ and (ii) unconditionally. The method used in the conditional case seems applicable to maxima of a reasonably large class of random fields. For the unconditional probability a more special argument is used, and more precise results obtained.

#### Article information

Source
Ann. Probab., Volume 16, Number 2 (1988), 487-501.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991769

Digital Object Identifier
doi:10.1214/aop/1176991769

Mathematical Reviews number (MathSciNet)
MR929059

Zentralblatt MATH identifier
0646.60032

JSTOR