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Strong Mixing Properties of Linear Stochastic Processes

K. C. Chanda
Journal of Applied Probability
Vol. 11, No. 2 (Jun., 1974), pp. 401-408
DOI: 10.2307/3212764
Stable URL: http://www.jstor.org/stable/3212764
Page Count: 8
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Strong Mixing Properties of Linear Stochastic Processes
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Abstract

Let {Zt; = 0, ± 1, ⋯} be a pure white noise process with $\gamma = E\{|Z_{1}|^{\delta}\} < \infty$ for some $\delta > 0$. Assume that the characteristic function (ch.f)φ0 of Z1 is Lebesgue-integrable over (- ∞, ∞). Let {gv;V = 0,1,2, ⋯, g0 = 1} be a sequence of real numbers such that $\Sigma_{v=0}^{\infty}v |g_{v}| ^{\lambda} < \infty$ where λ = δ(1+δ)-1. Define Xt = Σv=0 ∞gv Zt-v, where the identity is to be understood in the sense of convergence in distribution. Then {Xt;t=0,± 1, ⋯} is a strongly mixing stationary process in the sense that if Ma b(b≥ a) is the σ-field generated by the random variables (r.v.) Xa, ⋯, Xb then for any $A \in \mathcal{M}_{-\infty}^{0},\, B \in \mathcal{M}_{k}^{\infty}\, |P(AB)-P(A)P(B)|\, <\, Ma(k)$ where M is a finite positive constant which depends only on φ0 and $a(k) = \Sigma_{j=k}^{\infty}v |g_{v}|^{\lambda},\, \lambda = \delta/(1 + \delta),\, (a(k)\downarrow 0\, {as}\, k \rightarrow \infty)$.

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