Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on \(p\)-adic vector spaces
Let \(V\) be a finite-dimensional vector space over the field \(\mathbb Q_p\) of \(p\)-adic numbers, and let \(\tau\) be an operator in \(GL(V)\). A probability measure \(\mu\) on \(V\) is called \(\tau\)-decomposable and denoted \(\mu \in \tilde{L}_0(\tau )\), if \(\mu =\tau (\mu )*\rho_0\) for some probability measure \(\rho_0\). Then the authors introduce the classes \[ \tilde{L}_m(\tau )=\{ \mu=\tau (\mu )*\rho_m,\rho_m\in \tilde{L}_{m-1}(\tau )\},\quad m\geq 1, \] \[ \tilde{L}_\infty (\tau )=\bigcap\limits_m\tilde{L}_m(\tau ). \] Their properties are studied in detail, as well as properties of classes constructed in a similar way, with an infinitely divisible or embeddable measure \(\rho_0\). For similar constructions in the framework of real vector spaces, see [\textit{M. Maejima, K. Sato} and \textit{T. Watanabe}, Tokyo J. Math. 22, No.~2, 473--509 (1999; Zbl 0947.60010); Tokyo J. Math. 23, No.~1, 235--253 (2000; Zbl 0985.60014)].
scientific article; zbMATH DE number 5189631