Kouritzin, Michael A. - In: Stochastic Processes and their Applications 60 (1995) 2, pp. 343-353
Suppose {[var epsilon]k, -[infinity] < k < [infinity]} is an independent, not necessarily identically distributed sequence of random variables, and {cj}[infinity]j=0, {dj}[infinity]j=0 are sequences of real numbers such that [Sigma]jc2j < [infinity], [Sigma]jd2j < [infinity]. Then, under appropriate moment conditions on {[var epsilon]k, -[infinity] < k < [infinity]}, yk [triangle, equals][Sigma][infinity]j=0cj[var epsilon]k-j, zk [triangle, equals] [Sigma][infinity]j=0dj[var epsilon]k-j exist almost surely and in 4 and the question of Gaussian approximation to S[t][triangle, equals][Sigma][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 [var epsilon]k for "time" k <= 0, weaken the stationarity assumptions on {[var epsilon]k, -[infinity] < k < [infinity]}, and improve the summability conditions on {cj}[infinity]j=0, {dj}[infinity]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.