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

Abstract

Suppose { ε k , −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and { c j } ∞ j=0 , { d j } ∞ j=0 are sequences of real numbers such that Σ j c 2 j < ∞, Σ j d 2 j < ∞. Then, under appropriate moment conditions on { ε k , −∞ < k < ∞}, y k ≜Σ ∞ j=0c jε k-j , z k ≜ Σ ∞ j=0d jε k-j exist almost surely and in L 4 and the question of Gaussian approximation to S [t]≜Σ [t] k=1 (y k z k − E{y k z k}) 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 ε k for “time” k ≤ 0, weaken the stationarity assumptions on { ε k , −∞ < k < ∞}, and improve the summability conditions on { c j } ∞ j=0 , { d j } ∞ 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 d j = c j- m for j ≥ m and otherwise 0), quadratic forms, Whittle's and Hosoya's estimates, adaptive filtering and stochastic approximation.