We observe that the function Fα(x) = (1+αx α)e−x α is completely monotone iff α ≤ α0 for some α0 ∈ ]2/3, 3/4[. This property is equivalent to the unimodality of the inverse positive α-stable law. The random variable associated with Fα appears then in two different factorizations of the positive α-stable distribution. Furthermore, it is infinitely divisible iff α ≤ α1 for some α1 ∈ ]2/3, α0[ and...

The article provides an historical survey of the early contributions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Lévy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian duri...

Abstract: We extend the boson process first to a large class of Cox processes and second an even larger class of infinitely divisible point processes. Density and moment results are studied in detail. These results are obtained in closed form as weighted permanents, so the extension is called a permanent process. Temporal extensions and a particularly tractable case of the permanent process are...

This paper proposes to unify fading distributions by modeling the magnitude-squared of the instantaneous channel gain as an infinitely divisible random variable. A random variable is said to be infinitely divisible, if it can be written as a sum of n ≥ 1 independent and identically distributed random variables, for each n. Infinitely divisible random variables have many interesting mathematical...

The classical duel is a zero-sum game of two players of the following type. The players have certain resources and use them during a given time interval with the goal of achieving success. Use of the resource γ at the moment t leads to success with the probability depending on the amount of resource γ and the time t only (it is usually assumed that the probability of success increases with time...

The Brownian loop soup introduced in [3] is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, M...

We give a brief introduction to the class of stochastic processes known as Lévy processes, concentrating principally on their relation with infinitely divisible distributions and the Lévy-Itô decomposition.

It is known that the so-called Bercovici-Pata bijection can be explained in terms of certain Hermitian random matrix ensembles (Md)d≥1 whose asymptotic spectral distributions are free infinitely divisible. We investigate Hermitian Lévy processes with jumps of rank one associated to these random matrix ensembles introduced in [6] and [10]. A sample path approximation by covariation processes for...

In this paper we study differentiability and semismoothness properties of functions defined as integrals of parameterized functions. We also discuss applications of the developed theory to the problems of shape-preserving interpolation, option pricing and semi-infinite programming.