Open Access
February 2017 The impact of the diagonals of polynomial forms on limit theorems with long memory
Shuyang Bai, Murad S. Taqqu
Bernoulli 23(1): 710-742 (February 2017). DOI: 10.3150/15-BEJ697


We start with an i.i.d. sequence and consider the product of two polynomial-forms moving averages based on that sequence. The coefficients of the polynomial forms are asymptotically slowly decaying homogeneous functions so that these processes have long memory. The product of these two polynomial forms is a stationary nonlinear process. Our goal is to obtain limit theorems for the normalized sums of this product process in three cases: exclusion of the diagonal terms of the polynomial form, inclusion, or the mixed case (one polynomial form excludes the diagonals while the other one includes them). In any one of these cases, if the product has long memory, then the limits are given by Wiener chaos. But the limits in each of the cases are quite different. If the diagonals are excluded, then the limit is expressed as in the product formula of two Wiener–Itô integrals. When the diagonals are included, the limit stochastic integrals are typically due to a single factor of the product, namely the one with the strongest memory. In the mixed case, the limit stochastic integral is due to the polynomial form without the diagonals irrespective of the strength of the memory.


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Shuyang Bai. Murad S. Taqqu. "The impact of the diagonals of polynomial forms on limit theorems with long memory." Bernoulli 23 (1) 710 - 742, February 2017.


Received: 1 March 2014; Revised: 1 November 2014; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1360.60048
MathSciNet: MR3556790
Digital Object Identifier: 10.3150/15-BEJ697

Keywords: diagonals , long memory , noncentral limit theorem , Self-similar processes , Volterra , Wiener

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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