Weak Lévy--Khintchine Representation for Weak Infinite Divisibility

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, 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.

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

“…which shows that the measure Exp ⋄ (aλ) is infinitely divisible in the sense of generalized convolution ⋄. More about ⋄-infinite divisibility of measures (called sometimes also infinite decomposability) one can find in [16,6].…”

“…These formulas can be used in calculating the measure Exp ⊗ω 3 (cδ 1 ). However in [6], using much simpler method, it was shown that for any c > 0…”

“…As far as we know the generalized Poisson processes in the sense of generalized convolution have not been studied yet, but Markov processes which can be considered as additive processes in the sense of generalized convolutions have been considered in [2,14,22]. Properties of ⋄-generalized Poisson distribution were studied in [6,16]. Basic properties and a list of examples of generalized convolutions is given in Section 2.…”

Kendall random walk,Williamson transform, and the corresponding Wiener–Hopf factorization

“…Notice that properties of Kendall convolution together with the one described in the last proposition show that the Kendall convolution is an example of generalized convolutions in the sense defined by K. Urbanik. More about generalized convolutions one can find in [2,5,6,7,8,9,10,11].…”

Lévy processes and stochastic integrals in the sense of generalized convolutions

“…There are measures λ and weakly stable measures µ such that µ • λ is infinitely divisible and λ is not µ-weakly infinitely divisible. Counterexamples are known even for µ symmetric Gaussian and symmetric stable measures µ (see Example 2 in [10]). Special properties of infinitely divisible substable distributions are discussed in [17,25,26].…”