scholarly journals On a transformation of the weighted compound Poisson process

1971 ◽  
Vol 6 (1) ◽  
pp. 42-46 ◽  
Author(s):  
Hans Bühlmann ◽  
Roberto Buzzi
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Characteristic Functions ◽  
Compound Poisson ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Poisson Variable ◽  
Image Position ◽  
Independent Identically Distributed

We are using the following terminology—essentially following Feller:a) Compound Poisson VariableThis is a random variable where X1, X2, … Xn, … independent, identically distributed (X0 = o) and N a Poisson counting variablehence(common) distribution function of the Xj with j ≠ 0 or in the language of characteristic functionsb) Weighted Compound Poisson VariableThis is a random variable Z obtained from a class of Compound Poisson Variables by weighting over λ with a weight function S(λ)henceor in the language of characteristic functionsLet [Z(t); t ≥ o] be a homogeneous Weighted Compound Poisson Process. The characteristic function at the time epoch t reads thenIt is most remarkable that in many instances φt(u) can be represented as a (non weighted) Compound Poisson Variable. Our main result is given as a theorem.

An extension of a lemma of Wald

1966 ◽  
Vol 3 (01) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y 1, y 2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

scholarly journals A Generalized Model for the Risk Process and its Application to a Tentative Evaluation of Outstanding Liabilities

1965 ◽  
Vol 3 (3) ◽  
pp. 215-238 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Weak Sense ◽  
Risk Process ◽  
Distribution Functions ◽  
Wide Sense ◽  
Risk Distribution ◽  
Image Position ◽  
The Right

A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.

An extension of a lemma of Wald

1966 ◽  
Vol 3 (1) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y1, y2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

scholarly journals On the Numerical Calculation of the Distribution Functions Defining Some Compound Poisson Processes

1963 ◽  
Vol 3 (1) ◽  
pp. 20-42 ◽  
Author(s):  
Carl Philipson
Keyword(s):  
Probability Distribution ◽  
Poisson Process ◽  
Random Function ◽  
Compound Poisson Process ◽  
Distribution Functions ◽  
Wide Sense ◽  
Stieltjes Integral ◽  
Compound Poisson ◽  
Risk Distribution ◽  
Image Position

1. The comfound Poisson process in the wide sense is defined as a process for which the probability distribution of the number i of changes in the random function attached to the process, while the parameter passes from o to a fixed value τ of the parameter measured on a suitable scale, is given by the Laplace-Stieltjes integral where U(ν, τ) for a fixed value of τ defines the distribution of ν. U(ν, τ) is called the risk distribution and is either τ-independent or, dependent on ν, τ.2. The compound Poisson process in the narrow sense is defined as a process for which the probability distribution of the number of changes can be written in the form of (I) with a τ-independent risk distribution.In their general form these processes have been analyzed by Ove Lundberg (1940). For such processes the following relation holds for the probability of i changes in the interval ο to τ, P̅i (τ) say this relation does not hold for processes with τ-dependent risk distribution. Hofmann (1955) has introduced a sub-set of the processes concerned in this section for which the probability for non-occurrence of a change in the interval o to τ is defined as a solution of the differential equation and ϰ ≥ o; the solutions may be written in the form where η is independent of and of two alternative forms one for ϰ = I and one for other values of ϰ. The probabilities for i changes in the interval o to τ in the processes defined by the solutions of Hofmann's equation are derived by Leibniz's formula, and are designated by and, in this paper, called Hofmann probabilities.

Approximations for the single-product production-inventory problem with compound Poisson demand and service-level constraints

1984 ◽  
Vol 16 (2) ◽  
pp. 378-401 ◽  
Author(s):  
A. G. De kok ◽  
H. C. Tijms ◽  
F. A. Van der Duyn Schouten
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Service Level ◽  
Inventory Level ◽  
Inventory Problem ◽  
Single Product ◽  
Compound Poisson ◽  
Poisson Demand ◽  
Production Inventory ◽  
Random Fluctuations

We consider a production-inventory problem in which the production rate can be continuously controlled in order to cope with random fluctuations in the demand. The demand process for a single product is a compound Poisson process. Excess demand is backlogged. Two production rates are available and the inventory level is continuously controlled by a switch-over rule characterized by two critical numbers. In accordance with common practice, we consider service measures such as the average number of stockouts per unit time and the fraction of demand to be met directly from stock on hand. The purpose of the paper is to derive practically useful approximations for the switch-over levels of the control rule such that a pre-specified value of the service level is achieved.

Sign in / Sign up

Export Citation Format

Share Document