Extending the Thorin class

Abstract

We prove that the Thorin class ${T_\kappa }\left( {{\mathbb{R}_{+} }} \right),\kappa > 0$, is the minimal class of probability distributions on ${\mathbb{R}_{+} }$ that is closed under convolutions and weak limits and contains the Tweedie distributions Tw_(κ+1)/κ (μ, λ), μ > 0, λ > 0.

We characterize the Poisson transform ${T_\kappa }\left( {{\mathbb{Z}_{+} }} \right)$ of ${T_\kappa }\left( {{\mathbb{R}_{+} }} \right)$. Continuing the relationship of Tweedie and Thorin distributions and using the canonical polar characteristics of infinitely divisible distributions on ${\mathbb{R}^d}$, we define and characterize analytically the extended classes of Thorin distributions ${T_\kappa }\left( {{\mathbb{R}^d}} \right)$, −1 ⩽ κ ⩽ ∞. As an example, we describe multivariate Tweedie distributions.