# Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

## Details

Serval ID

serval:BIB_DA0CEF069158

Type

**Article**: article from journal or magazin.

Collection

Publications

Institution

Title

Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Journal

Methodology And Computing In Applied Probability

ISSN

1387-5841

Publication state

Published

Issued date

2003

Peer-reviewed

Oui

Volume

5

Number

3

Pages

271-287

Language

english

Abstract

Let us consider a signal-plus-noise model

yh(z) + B-0(z), z epsilon [0, 1],

where gamma > 0, h : [0, 1] --> R, and B-0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for gamma --> infinity, that is

P(sup(zepsilon(0,1)) w(z) (gammah(z) + B-0(z)) > c), for gamma --> infinity, (1)

where w : [0, 1] --> [0, infinity) is a weight function and c > 0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H-o : h equivalent to 0 against the altemative K : h > 0 in the signal-plus-noise model.

yh(z) + B-0(z), z epsilon [0, 1],

where gamma > 0, h : [0, 1] --> R, and B-0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for gamma --> infinity, that is

P(sup(zepsilon(0,1)) w(z) (gammah(z) + B-0(z)) > c), for gamma --> infinity, (1)

where w : [0, 1] --> [0, infinity) is a weight function and c > 0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H-o : h equivalent to 0 against the altemative K : h > 0 in the signal-plus-noise model.

Keywords

Brownian bridge with trend, Boundary crossing probability, Asymptotic results, Large deviations, Signal-plus-noise model, Tests of Kolmogorov type

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Create date

03/09/2010 11:15

Last modification date

20/08/2019 15:59