scholarly journals The closure of a local subexponential distribution class under convolution roots, with applications to the compound Poisson process

2005 ◽  
Vol 42 (04) ◽  
pp. 1194-1203 ◽  
Author(s):  
Yuebao Wang ◽  
Dongya Cheng ◽  
Kaiyong Wang
Keyword(s):  
Finite Number ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Subexponential Distributions ◽  
Infinitely Divisible ◽  
Poisson Random Variable ◽  
Infinitely Divisible Laws ◽  
Geometric Random Variable ◽  
Distribution Class

Let denote the class of local subexponential distributions and F ∗ν the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν ≡ n for some n ≥ 2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).

scholarly journals The closure of a local subexponential distribution class under convolution roots, with applications to the compound Poisson process

2005 ◽  
Vol 42 (4) ◽  
pp. 1194-1203 ◽  
Author(s):  
Yuebao Wang ◽  
Dongya Cheng ◽  
Kaiyong Wang
Keyword(s):  
Finite Number ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Random Variable ◽  
Subexponential Distributions ◽  
Infinitely Divisible ◽  
Poisson Random Variable ◽  
Infinitely Divisible Laws ◽  
Geometric Random Variable ◽  
Distribution Class

Let denote the class of local subexponential distributions and F∗ν the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν ≡ n for some n ≥ 2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

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.

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.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (04) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (4) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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