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On a Szegö type limit theorem and the asymptotic theory of random sums, integrals and quadratic forms. (English) Zbl 1113.60024

Bertail, Patrice (ed.) et al., Dependence in probability and statistics. New York, NY: Springer (ISBN 0-387-31741-4/pbk). Lecture Notes in Statistics 187, 259-286 (2006).
It is important to develop models and methods which replace the assumption of a one-dimensional discrete time index in one-dimensional time series with long memory with that of a multi-dimensional continuous one. This paper is motivated by the attempt to extend to the case of multi-dimensional continuous indices certain central limit theorems of Giraitis and Surgailis, Fox and Taqqu and Giraitis and Taqqu, in which the approach of these papers reduced the central limit theorems considered to an application of three analytical tools: (1) diagram formula for computing moments/cumulants of Wich products, (2) generalization of Hölder-Young inequality, which has become more recently known as the Hölder-Brascamp-Lieb-Barthe inequality, (3) some generalizations of a Grenander-Szegö theorem on the trace of products of Toeplitz matrices. The authors review throughout the paper the one-dimensional discrete results in previous papers and formulate the results using a unifying measure theoretic notation, which includes the discrete one-dimensional setup and the continuous multi-dimensional one.
For the entire collection see [Zbl 1092.60002].

MSC:

60F05 Central limit and other weak theorems
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