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

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385700010191How to cite this article: Eric Slud and Daniel Chambers (1996). Mixing for stationary processes with nite-order multiple Wiener-Itô integral representation.Abstract. Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Ito 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 Y,, for sums of MWIs of order < 3, and for a large class of square-integrable infinite sums Y, of MWIs, mixing holds if and only if K, 2 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.