We consider centered compound Poisson processes with finite variance, discretely observed over [0, T ] and let the sampling rate ∆ = ∆T → ∞ as T → ∞. From the central limit theorem, the law of each increment converges to a Gaussian variable. Then, it should not be possible to estimate more than one parameter at the limit. First, from the study of a parametric example we identify two regimes for...

For the stationary loss systems M/M/m/K and GI/M/m/K, we study two quantities: the number of lost customers during the time interval (0, t] (the first system only), and the number of lost customers among the first n customers to arrive (both systems). We derive explicit bounds for the total variation distances between the distributions of these quantities and compound Poisson-geometric distribu...

This is a collection of corrections and additions to the book, and references to subsequent work. Perhaps the main development since going to press has been continued work by Barbour et al and by Arratia et al, referenced below, on using Stein's method to get explicit bounds for Poisson approximation. However, I have not yet seen any convincing implementation of the Compound Poisson case. Also ...

We consider a distribution system with a central warehouse and multiple retailers. The warehouse orders from an outside supplier and replenishes the retailers which in turn satisfy customer demand. The retailers are nonidentical, and their demand processes are independent compound Poisson. There are economies of scale in inventory replenishment, which is controlled by an echelon-stock, batch-tr...

Abstract. This paper begins by defining the basic aspects of probability, including the probability space and random variables. We explore the types and qualities of random variables, as well as di↵erent distributions and their properties, with the goal of building the Poisson Process, and finally the Compound Poisson Process. This is an important counting process that can be applied to many in...

We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

The Tweedie compound Poisson distribution is a mixture of a degenerate distribution at the origin and a continuous distribution on the positive real line. It has been applied in a wide range of fields in which continuous data with exact zeros regularly arise. Nevertheless, statistical inference based on full likelihood and Bayesian methods is not available in most statistical software, largely ...

and key words Thirty years ago, Bernard Picinbono and his colleagues carefully addressed an important problem : how an optical field is converted into a sequence of photoelectrons upon detection . Their choice of problem could not have been better, nor thei r timing more judicious. In a paper entitled "Photoelectron Shot Noise", published in the Journal of Mathematical Physics in 1970, when qua...