Strong approximation for cross-covariances of linear variables with long-range dependence.

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  • Suppose {εk, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {cj}∞j=0, {dj}∞j=0 are sequences of real numbers such that Σjc2j < ∞, Σjd2j < ∞. Then, under appropriate moment conditions on {εk, −∞ < k < ∞}, View the MathML source, View the MathML source exist almost surely and in View the MathML source4 and the question of Gaussian approximation to View the MathML source becomes of interest. Prior to this work several related central limit theorems and a weak invariance principle were proven under stationary assumptions. In this note, we demonstrate that an almost sure invariance principle for S[t], with error bound sharp enough to imply a weak invariance principle, a functional law of the iterated logarithm, and even upper and lower class results, also exists. Moreover, we remove virtually all constraints on εk for “time” k ≤ 0, weaken the stationarity assumptions on {εk, −∞ < k < ∞}, and improve the summability conditions on {cj}∞j=0, {dj}∞j=0 as compared to the existing weak invariance principle. Applications relevant to this work include normal approximation and almost sure fluctuation results in sample covariances (let dj = cj-m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.

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    © 1995 Michael Kouritzin. This version of this article is open access and can be downloaded and shared. The original author(s) and source must be cited. Non-commercial use only.
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    • Michael A. Kouritzin, Strong approximation for cross-covariances of linear variables with long-range dependence, Stochastic Processes and their Applications, Volume 60, Issue 2, December 1995, Pages 343-353, ISSN 0304-4149,