Abstract. For a random vector X with a fixed distribution µ we construct a class of distributions M(µ) = {µ • λ : λ ∈ P}, which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions µ for which M(µ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ 1 , Θ 2 independent of X, X there exists a random variable Θ independent of X such thatWe show that for every X this property is equivalent to the following condition:This condition reminds the characterizing condition for symmetric stable random vectors, except that Θ is here a random variable, instead of a constant. The above problem has a direct connection with the concept of generalized convolutions and with the characterization of the extreme points for the set of pseudo-isotropic distributions.
Introduction. Let E be a separable real Banach space. By P(E)we denote the set of all Borel probability measures on E. For E = R we will use the simplified notation P(R) = P, and the set of all probability measures on [0, ∞) will be denoted by P + . For every a ∈ R and every probability measure µ, we define the rescaling operator µ → T a µ by the formula (T a µ)(A) = µ(A/a) when a = 0, and T 0 (µ) = δ 0 . This means that T a µ is the distribution of the random vector aX if µ is the distribution of the vector X. For every µ ∈ P(E) and λ ∈ P we define a scale mixture µ • λ
In this paper, we present a comprehensive theory of generalized and weak generalized convolutions, illustrate it by a large number of examples, and discuss the related infinitely divisible distributions. We consider Lévy and additive process with respect to generalized and weak generalized convolutions as certain Markov processes, and then study stochastic integrals with respect to such processes. We introduce the representability property of weak generalized convolutions. Under this property and the related weak summability, a stochastic integral with respect to random measures related to such convolutions is constructed.
The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.
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, where X ′ is an independent copy of X and Θ is independent of X. This is equivalent (see [12]) 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 weak generalized convolution of measures defined by the formulaif the equation ( * ) holds for X, Q 1 , Q 2 , 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 analog of the Lévy-Khintchine representation theorem for ⊗ µ -infinitely divisible distributions.
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