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What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?

In a similar way let us consider a stationary vector valued Gaussian process indexed by an infinite discrete abelian group with mean zero. What is the condition of ergodicity of such a process in terms of its spectral measure?

I apologize if this question is not of mathoverflow standard. I am not an expert of this subject. But I need to learn some points in this matter urgently. So can someone please suggest me some reference books on stationary stochastic process, and stationary Gaussian process, which starts from basic theorems and goes (perhaps) upto the state of art on this subject?

Advanced thanks for any suggestion, comment, and etc.

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If $\mu$ is the spectral measure of the Gaussian process, then the maximal spectral type of the shift map on the distribution of the process is $\exp(\mu):=\sum_{k=1}^n\mu^{\ast k}/k!$ where $\mu^{\ast k}$ is the $k$ fold convolution of $\mu$ (this comes from the Fock spaces decomposition). Using this characterisation one can show that a Gaussian process is ergodic if and only if the spectral measure has no atoms and thus ergodicity and weak mixing are equivalent for Gaussian processes. The process is mixing if and only if the spectral measure is a Rajchman measure, meaning that its Fourier coefficients decay to $0$ as $n\to\infty$. For more see the survey of Katok and Thouvenot on spectral properties of dynamical systems (it's a chapter in one of the Encyclopedia's edited by Katok and Hasselblatt).

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  • $\begingroup$ Thank you very much for your answer. I can see one way of the result but for the other way I am searching for Katok et al. One more thing. What happens for vector valued process? This must have been solved by now. Can you please give me some information about this? Please see the modified question above. Thank you once again. $\endgroup$
    – RSG
    Jun 18 '16 at 4:23
  • $\begingroup$ I think the paper "Maruyama, G. Infinitely divisible processes" will give you an answer for the multidimensional case (you need to disregard the Poisson integral part) $\endgroup$
    – user78465
    Jun 20 '16 at 22:57

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