NECESSARY AND SUFFICIENT CONDITIONS FOR A 2ND-ORDER WIENER-ITO INTEGRAL PROCESS TO BE MIXING

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:45

Article

关键词: MIXING;    STATIONARY PROCESS;    GAUSSIAN PROCESS;    MULTIPLE WIENER-ITO INTEGRAL;   

DOI  :  10.1016/0304-4149(93)90068-F

来源: Elsevier

【 摘 要 】

Let (X(s), s is-an-element-of Z) be a stationary Gaussian process with spectral measure sigma, time-shift operator U, and the associated pth order multiple Wiener-Ito integrals, I(p), p = 1, 2,..., defined on their domains L2(sigma(p), sym). Let f is-an-element-of L2(sigma(p), sym). We give a necessary and sufficient spectral condition for the stationary process (U(s)(I(p)f), s is-an-element-of Z) to be mixing in the case p = 2; a simplified sufficient condition is given for f of the form f = g1 x h1 + g2 x h2 +...+g(n) x h(n), where g(i), h(i) is-an-element-of L2(sigma1, sym). Similar results are obtained in the case p = 4. A necessary and sufficient spectral condition is given for (U(s)(I(p)(h x...x h)), s is-an-element-of Z) to be mixing, for any p greater-than-or-equal-to 1 and h is-an-element-of L2(sigma1, sym). An example of a non-mixing stationary Gaussian process with a mixing factor process is given.

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