Abstract
We investigate the joint weak convergence (f.d.d. and functional) of the vector-valued process (U (1)n (τ), U (2)n (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 (1)n (τ), U (2)n (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.
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This work supported by the joint Lithuania-French research program Gilibert.
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Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 479–500, October–December, 2005.
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Bruzaite, K., Vaiciulis, M. Asymptotic Independence of Distant Partial Sums of Linear Processes. Lith Math J 45, 387–404 (2005). https://doi.org/10.1007/s10986-006-0003-5
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DOI: https://doi.org/10.1007/s10986-006-0003-5