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

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 < ∞}, yk ≜Σ∞j=0cjεk-j, zk ≜ Σ∞j=0djεk-j exist almost surely and in L4 and the question of Gaussian approximation to S[t]≜Σ[t]k=1 (yk zk − E{yk zk}) 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, Whittleʹs and Hosoyaʹs estimates, adaptive filtering and stochastic approximation.