Weak stability and generalized weak convolution for random vectors and stochastic processes

Abstract

A random vector X is weakly stable iff for all a, b ∈ R there exists a random variable Θ such that aX + bX ′ d = XΘ. This is equivalent (see [11]) with the condition that for all random variables Q 1 , Q 2 there exists a random variable Θ such thatwhere X, X ′ , Q 1 , Q 2 , Θ are independent. In this paper we define generalized convolution of measures defined by the formulaif the equation ( * ) holds for X, Q 1 , Q 2 , Θ and µ = L(Θ). We study here basic properties of this convolution, basic properties of ⊕µ-infinitely divisible distributions, ⊕µ-stable distributions and give a series of examples.

Classical definitions of the Poisson process do not coincide in the case of generalized convolutions

“…The generalized convolutions on the set P + of probability measures on the Borel subsets of the positive half line were defined by Urbanik (see [16]). It is also possible to consider more general binary operation also called generalized convolution defined on the set P of all probability measures on the Borel subsets of the real line R (see [3,12,13]) or generalized convolutions on the set P s of all symmetric probability measures on R. When our result holds in every of these cases we say that considered measures are living on K.…”

On $\mathbf R^+$-weakly stable distribution

“…Weak stability is also the subject of many papers of Vol'kovich (see, e.g., [12]- [14]). Recently there appeared a paper written by Misiewicz (see [6]).…”

On analytical properties of generalized convolutions

“…Here, they were named weak-stable distributions. Moreover, Misiewicz [33] defined infinitely divisible measures with respect to the generalized weak convolution. Jasiulis and Misiewicz [14] gave the Lévy-Khintchine representation for an infinitely divisible measure and defined µ-stable measures in the sense of weak generalized convolution.…”

“…lutions concerning the quasi-stable functions and the related B-stable distributions was proposed in [22], [23] and further developed in [47] (see also [64]). Generalizations of this approach make it possible to create and design new probability models, which were introduced by Misiewicz and her colleagues [13], [33], [34]. The V -infinitely divisible measures introduced by Volkovich [68], [71], [73] were prompted by the general convolutions concept as an attempt to obtain a generalization for the whole real line.…”

“…Misiewicz, Oleszkiewicz and Urbanik [34] provided a full characterization of weakly stable distributions with a non-trivial discrete part and also substantially contributed to characterizing weakly stable distributions in a general case. Convolutions produced in this way were studied by Misiewicz [33], while Jasiulis and Misiewicz [13] described the relation between weakly stable and pseudo-isotropic distributions.…”