WEAK LÉVY-KHINTCHINE REPRESENTATION FOR WEAK INFINITE DIVISIBILITY.

A random vector X is weakly stable if and only if for all a, b ∈ ℝ there exists a random variable Θ such that aX + bX' ... XΘ, where X' is an independent copy of X and Θ is independent of X. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195-213]) to the condition that for all random variables Q₁,Q₂ there exists a random variable Θ such that XQ₁ +X'Q₂ ... XΘ, where X,X',Q₁,Q₂,Θ are independent. In this paper we define a weak generalized convolution of measures defined by the formula L(Q₁) ⊗μ L(Q₂) = L(Θ), if the former equation holds for X,Q₁,Q₂,Θ and μ = L(X). We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this convolution. The main result of this paper is the analogue of the Lévy-Khintchine representation theorem for ⊗μ-infinitely divisible distributions. [ABSTRACT FROM AUTHOR]