Asymptotic Independence of Distant Partial Sums of Linear Processes

Abstract

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 $$U_n^{(1)} (\tau )\;: = \;A_n^{ - 1} \;\Sigma _{t = 1}^{[n\tau ]} \;X_t $$ and $$U_n^{(2)} (\tau )\;: = \;A_n^{ - 1} \;\Sigma _{t = 1}^{[n\tau ]} \;X_{t + m} $$ 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.