Multiple stable integrals of Banach-valued functions
This paper is devoted to a study of the multiple stable integrals of Banach-valued functions. Let M be a symmetric \(\alpha\)-stable random measure and S be a separable Banach space of Rademacher-type \(p\geq 1\), with \(p>\alpha\). If \(f: {\mathbb{R}}^ n\to S\) is a symmetric deterministic function vanishing on the diagonals then a sufficient condition is given for the existence of \(I_ n(f)\), a multiple integral of order n with respect to M. Moreover, the tail behaviour of the distribution function of the multiple integral is computed, namely, a formula for \(P(\| I_ n(f)\| >\epsilon)\) is given. Some applications are also presented. The paper extends, to the case of Banach-valued integrands, results of the first author and \textit{J. Szulga} [Ann. Probab. 17, No.4, 1503-1520 (1989; Zbl 0689.60051)].
scientific article; zbMATH DE number 4149282