## The Annals of Probability

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

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

#### 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**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60G60: Random fields 60K05: Renewal theory 62N10

**Keywords**

Large deviations random field CUSUM test

#### Citation

Siegmund, David. Approximate Tail Probabilities for the Maxima of Some Random Fields. Ann. Probab. 16 (1988), no. 2, 487--501. doi:10.1214/aop/1176991769. https://projecteuclid.org/euclid.aop/1176991769