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April 2019 Change-point detection for Lévy processes
José E. Figueroa-López, Sveinn Ólafsson
Ann. Appl. Probab. 29(2): 717-738 (April 2019). DOI: 10.1214/17-AAP1368

Abstract

Since the work of Page in the 1950s, the problem of detecting an abrupt change in the distribution of stochastic processes has received a great deal of attention. In particular, a deep connection has been established between Lorden’s minimax approach to change-point detection and the widely used CUSUM procedure, first for discrete-time processes, and subsequently for some of their continuous-time counterparts. However, results for processes with jumps are still scarce, while the practical importance of such processes has escalated since the turn of the century. In this work, we consider the problem of detecting a change in the distribution of continuous-time processes with independent and stationary increments, that is, Lévy processes, and our main result shows that CUSUM is indeed optimal in Lorden’s sense. This is the most natural continuous-time analogue of the seminal work of Moustakides [Ann. Statist. 14 (1986) 1379–1387] for sequentially observed random variables that are assumed to be i.i.d. before and after the change-point. From a practical perspective, the approach we adopt is appealing as it consists in approximating the continuous-time problem by a suitable sequence of change-point problems with equispaced sampling points, and for which a CUSUM procedure is shown to be optimal.

Citation

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José E. Figueroa-López. Sveinn Ólafsson. "Change-point detection for Lévy processes." Ann. Appl. Probab. 29 (2) 717 - 738, April 2019. https://doi.org/10.1214/17-AAP1368

Information

Received: 1 November 2016; Revised: 1 June 2017; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047436
MathSciNet: MR3910015
Digital Object Identifier: 10.1214/17-AAP1368

Subjects:
Primary: 60G51, 62L10
Secondary: 60G40, 62C20

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2019
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