Zones of attraction of self-similar multiple integrals

TLDR: In this article, Surgailis et al. considered the multidimensional case of convergence to self-similar fields and gave a short survey of the separate sections of the paper, including the connection of this theorem with the result of Dobrushin-Major [8] and some similar questions.

Abstract: ZONES OF ATTRACTION OF SELF-SIMILAR MULTIPLE INTEGRALS D. Surgailis UDC 519.21 The goal of this paper is the proof of the theorem announced in [20]. Here we consider the "multidimensional case," i.e., convergence to self-similar fields. We give a short survey of the separate sections. In Sec. 1 the concepts needed are formulated as well as the basic result of the paper (Theorem i). The connection of this theorem with the result of Dobrushin-Major [8] is discussed as well as some similar questions. In Sec. 2 the term making the basic contribution to the distribution of the sums considered is isolated. Here we explain the idea of the following proof, which is broken up into several lemmas, and their formulations are given. The proofs of these lemmas (except for Lemma 5) are carried out to section 3. In section 4 there is proved a lemma (Lemma 1 of [20]) on the convergence with respect to distribution of "discrete multiple integrals" to "continuous" integrals of Ito-- Wiener. With the help of this lemma, the remaining Lemma 5 is proved. i. Notation for what follows: R d is d-dimensional Euclidean space, x.y, ]xl are respectively the scalar product and norm in R d, Z d is the integer-valued d-dimensional lattice. We shall write