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.

Abstract. The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best