The class of type G distributions on \(\mathbb{R}^d\) and related subclasses of infinitely divisible distributions (Q2732342)

An infinitely divisible distribution of type G (of type TG) on \( \mathbb{R}^{d}\) is defined as a distribution with a canonical measure \(\nu\) of the form NEWLINE\[NEWLINE\nu(A)=\text{{E}}[\nu_{0}(Z^{-1}A)],\tag{1}NEWLINE\]NEWLINE where \(Z\) is a standard-normal random variable and \(\nu_{0}\) is measure on the Borel subsets of \(\mathbb{R}^{d}\). Starting from a TG-distribution, an ordered class of such distributions is constructed by iterating the relation between (canonical) measures in (1). These nested classes of infinitely divisible distributions are connected with the nested classes of self-decomposable distributions introduced by Urbanik and by \textit{K.-i. Sato} [J. Multivariate Anal. 10, 207-232 (1980; Zbl 0425.60013)]. The theory developed gives rise to explicit examples of infinitely divisible distributions. Not surprisingly, on \(\mathbb{R}\) the TG-distributions are related to variance mixtures of normal distributions.