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On a time deformation reducing nonstationary stochastic processes to local stationarity

Part of: Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Marc G. Genton  and
Olivier Perrin
Marc G. Genton*
Affiliation:
North Carolina State University
Olivier Perrin*
Affiliation:
Université Toulouse 1
*
∗ Postal address: Department of Statistics, North Carolina State University, Box 8203, Raleigh, NC 27695-8203, USA. Email address: genton@stat.ncsu.edu
∗∗ Postal address: GREMAQ, Université des Sciences Sociales, 21 Allée de Brienne, 31000 Toulouse, France. Email address: perrin@cict.fr
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Abstract

A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.

Keywords

Exponentially convexlocal stationaritynonstationaritypositive-definite functionreducibilitystationarityseparable covariance functionstochastic processestime series

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 62M30: Spatial processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 1 , March 2004 , pp. 236 - 249
Copyright
Copyright © Applied Probability Trust 2004 

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