Asymptotic Independence of Distant Partial Sums of Linear Processes

We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) for τ ∈ [0, 1], where and are normalized partial-sum processes separated by a large lag m, m/n → ∞, and (X _t , t ∈ ℤ) is a stationary moving-average process with i.i.d. (or martingale-difference) innovations having finite variance. We consider the cases where (X _t ) is a process with long memory, short memory, or negative memory. We show that, in all these cases, as n → ∞ and m/n → ∞, the bivariate partial-sum process (U _n ⁽¹⁾ (τ), U _n ⁽²⁾ (τ)) tends to a bivariate fractional Brownian motion with independent components. The result is applied to prove the consistency of certain increment-type statistics in moving-average observations. This work supported by the joint Lithuania-French research program Gilibert. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005.