Convergence of Appell Polynomials of Long Range Dependent Moving Averages in Martingale Differences

Authors: Surgailis, D.1; Vaičiulis, M.2

Source: Acta Applicandae Mathematicae, Volume 58, Number 1-3, September 1999 , pp. 343-357(15)

Publisher: Springer

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Abstract:

We study limit distribution of partial sums S_N, k(t) = ∑_s = 1^[N t] A_k(X_s) of Appell polynomials of the long-range dependent moving average process X_t = ∑_i  t b_t − i _i, where {_i} is a strictly stationary and weakly dependent martingale difference sequence, and b_i ∼ i^d − 1 (0 < d < 1 / 2). We show that if k(1 - 2 d) < 1, then suitably normalized partial sums S_N, k(t) converge in distribution to the kth order Hermite process. This result generalizes the corresponding results of Surgailis, and Avram and Taqqu obtained in the case of the i.i.d. sequence {_i}.

Keywords: Appell polynomials; linear process in martingale differences; long memory; non-central limit theorem

Document Type: Regular Paper

Affiliations: 1: Institute of Mathematics and Informatics, Akademijos 4, LT-2600 Vilnius, and Department of Mathematics, Šiauliai University, Višinskio 25, LT-5400 Šiauliai, Lithuania. e-mail: sdonatas@ktl.mii.lt 2: Department of Mathematics, Šiauliai University, Višinskio 25, LT-5400 Šiauliai, Lithuania. e-mail: marius@takas.lt

Publication date: September 1, 1999

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