Completely operator-selfdecomposable distributions and operator-stable distributions

Abstract

Urbanik introduces in [16] and [17] the classes Lm and L∞ of distributions on R1 and finds relations with stable distributions. Kumar-Schreiber [6] and Thu [14] extend some of the results to distributions on Banach spaces. Sato [7] gives alternative definitions of the classes Lm and L∞ and studies their properties on Rd. Earlier Sharpe [12] began investigation of operator-stable distributions and, subsequently, Urbanik [15] considered the operator version of the class L on Rd. Jurek [3] generalizes some of Sato’s results [7] to the classes associated with one-parameter groups of linear operators in Banach spaces. Analogues of Urbanik’s classes Lm (or L∞) in the operator case are called multiply (or completely) operator-selfdecomposable. They are studied in relation with processes of Ornstein-Uhlenbeck type or with stochastic integrals based on processes with homogeneous independent increments (Wolfe [18], [19], Jurek-Vervaat [5], Jurek [2], [4], and Sato-Yamazato [9], [10]). The purpose of the present paper is to continue the preceding papers, to give explicit characterizations of completely operator-selfdecomposable distributions and operator-stable distributions on Rd, and to establish relations between the two classes. For this purpose we explore the connection of the structures of these classes with the Jordan decomposition of a basic operator Q.