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Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

Published online by Cambridge University Press:  14 October 2010

Eric Slud  and
Daniel Chambers
Eric Slud
Affiliation:
Mathematics Department, University of Maryland, College Park, MD 20742, USA
Daniel Chambers
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02167-3806, USA
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Abstract

Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 5 , October 1996 , pp. 1087 - 1100
Copyright
Copyright © Cambridge University Press 1996

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References

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