Moments of distributions attracted to operator-stable laws
Let \(\mu\) and \(\gamma\) be probability measures on a Euclidean space V. We say that \(\gamma\) belongs to the generalized domain of attraction of \(\mu\) if there are matrices \(A_ n\) on V and vectors \(a_ n\in V\) such that \[ (*)\quad A_ n\gamma^{*n}*\delta (a_ n)\quad \Rightarrow \quad \mu, \] as \(n\to \infty\) and ``\(\Rightarrow ''\) denotes weak convergence. In case (*), \(\mu\) is called an operator-stable measure. With \(\mu\) is associated an operator (matrix) B called an exponent (of \(\mu)\). The paper deals with estimations for \(\| A_ n\|\) and \(\| A_ n^{-1}\|\) in terms of eigenvalues of B (Theorem 1 and 2) and inestigates p-moments of \(\gamma\) (Theorems 3 and 4). The authors use ``the more convenient norm'' from \textit{W. N. Hudson}, \textit{Z. I. Jurek} and \textit{J. A. Veeh} in Ann. Probab. 14, 1014-1023 (1986; Zbl 0603.60011)], Proposition 2, while in the paper the reader is referred to a preprint.
scientific article; zbMATH DE number 4040936