A characterization of mixing processes of type G
This paper discusses extensions for strictly stationary processes of type G -- a large class of stochastic processes, all of whose finite-dimensional distributions (fdd's) are infinitely divisible, with special integral representation for the log of the joint moment generating function of fdd's -- of the well-known characterization that a Gaussian process \(X_t\) is mixing if and only if \(\text{Cov} (X_0, X_t)\to 0\) as \(t\to 0\). The extended characterization is framed in terms of the function \[ I_t (\theta_1, \theta_2)=- \ln E \exp (i(\theta_1 X_0+ \theta_2 X_t))+ \ln E \exp (i\theta_i X_0)+ \ln E \exp (i\theta_0 X_t). \] A known result of Maruyama states that the stationary infinitely divisible process \(X_t\) is mixing if and only if for all \(\theta_1\), \(\theta_2\), \(\lim_{t\to \infty} I_t (\theta_1. \theta_2) =0\). In particular, the codifference function \(I_t (1, -1)\) enters into the main result, that the stationary infinitely divisible process \(X_t\) of type G is mixing if and only if both \(I_t (1, -1)\) and \(I_t (1, 1)\) tend to 0 as \(t\to \infty\).
scientific article; zbMATH DE number 847506
scientific article