A Notion of α-Monotonicity with Generalized Multiplications

The multiplications of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97–118) are used to generalize the definition of -monotonicity of Olshen and Savage (1970, J. Appl. Probab., 7, 21–34) and Steutel (1988, Statist. Neerlandica, 42, 137–140) for distributions with support in Z ₊ and R ₊. Several characterizations are offered and a convolution property is established. Some relevant stability equations are solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the R ₊-case from those for the Z ₊-case.     

Thorin classes of Lévy processes and their transforms

We define and characterize Thorin classes {ie294-01}, of infinitely divisible distributions on R ₊. We investigate Poisson, Karlin, and Bessel transforms of Thorin classes and also consider extended Thorin classes {ie294-02}. Canonical representation and self-decomposability properties of Thorin subordinated Gaussian Lévy processes are discussed. As an example, a subordinated Cauchy process is considered in detail.     

On the Self-Decomposability of Euler's Gamma Function

Let be Euler's Gamma function. We prove that, for all 0, > 0, > 0, > 0, the function (( + iz)/() ^{i z)} , z R ¹, is a self-decomposable characteristic function from the Thorin class and derive its explicit canonical form. Similarly to [1], we also describe several classes of Lévy-type stochastic processes related to .