Rosenblatt showed that a stationary Gaussian random field is strongly mixing if it has a positive, continuous spectral density. In this article, spectral criteria are given for the rate of strong mixing in such a field.
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Let ${X_{mn}}$ be a second-order stationary random field on Z². Let ℳ(L) be the linear span of ${X_{mn}: m ≤ 0, n ∈ Z}$, and ℳ(R_N) the linear span of ${X_{mn}: m ≥ N, n ∈ Z}$. Spectral criteria are given for the condition $lim_{N→∞} c_N = 0$, where $c_N$ is the cosine of the angle between ℳ(L) and $ℳ(R_N)$.
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An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it}) = Φ(e^{is},e^{it})*Φ(e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $Λ = {(m,n) ∈ ℤ^2: m ≥ 1} ∪ {(0,n): n ≥ 0}$, and Φ is "outer" in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2(W)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators and other structures associated with W. The corresponding version of Szegö's infimum is given.
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