Approximations for compound Poisson and Pólya processes

1985 ◽  
Vol 17 (3) ◽  
pp. 623-637 ◽  
Author(s):  
Paul Embrechts ◽  
Jens L. Jensen ◽  
Makoto Maejima ◽  
J. L. Teugels
Keyword(s):  
Saddlepoint Approximation ◽  
Compound Poisson ◽  
Polya Process ◽  
Pólya Process

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.

Approximations for compound Poisson and Pólya processes

1985 ◽  
Vol 17 (03) ◽  
pp. 623-637 ◽  
Author(s):  
Paul Embrechts ◽  
Jens L. Jensen ◽  
Makoto Maejima ◽  
J. L. Teugels
Keyword(s):  
Saddlepoint Approximation ◽  
Compound Poisson ◽  
Polya Process ◽  
Pólya Process

Suppose Xi ≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t) k=1 X k> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

Characterization of the Generalized Pólya Process and its Applications

2014 ◽  
Vol 46 (4) ◽  
pp. 1148-1171 ◽  
Author(s):  
Ji Hwan Cha
Keyword(s):  
Joint Distribution ◽  
Arrival Times ◽  
Stochastic Intensity ◽  
Polya Process ◽  
New Type ◽  
Compound Process ◽  
Pólya Process

In this paper some important properties of the generalized Pólya process are derived and their applications are discussed. The generalized Pólya process is defined based on the stochastic intensity. By interpreting the defined stochastic intensity of the generalized Pólya process, the restarting property of the process is discussed. Based on the restarting property of the process, the joint distribution of the number of events is derived and the conditional joint distribution of the arrival times is also obtained. In addition, some properties of the compound process defined for the generalized Pólya process are derived. Furthermore, a new type of repair is defined based on the process and its application to the area of reliability is discussed. Several examples illustrating the applications of the obtained properties to various areas are suggested.

Univariate and multivariate stochastic comparisons and ageing properties of the generalized Pólya process

2018 ◽  
Vol 55 (1) ◽  
pp. 233-253 ◽  
Author(s):  
F. G. Badía ◽  
C. Sangüesa ◽  
Ji Hwan Cha
Keyword(s):  
Intensity Function ◽  
Stochastic Comparisons ◽  
Polya Process ◽  
Pólya Process ◽  
Intensity Functions ◽  
Birth Processes

Abstract In this work we consider the generalized Pólya process with baseline intensity function r and parameters α and β recently studied by Cha (2014). The aim of this work is to provide both univariate and multivariate stochastic comparisons between two generalized Pólya processes with different baseline intensity functions and the same parameters α and β for the epoch and inter-epoch times of the two processes. The comparisons are analogous to stochastic comparisons in Belzunce et al. (2001) for two nonhomogeneous Poisson or pure-birth processes with different intensity functions. Moreover, we study both univariate and multivariate ageing properties of the epoch and inter-epoch times of the generalized Pólya process.

ON A MULTIVARIATE GENERALIZED POLYA PROCESS WITHOUT REGULARITY PROPERTY

2019 ◽  
Vol 34 (4) ◽  
pp. 484-506
Author(s):  
Ji Hwan Cha ◽  
F.G. Badía
Keyword(s):  
Regularity Property ◽  
Dependence Structure ◽  
Counting Processes ◽  
Multiple Events ◽  
New Class ◽  
Different Types ◽  
Polya Process ◽  
Multivariate Dependence ◽  
Pólya Process ◽  
Stochastic Properties

Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.

Estimating the Pólya process

2016 ◽  
Vol 46 (19) ◽  
pp. 9397-9406 ◽  
Author(s):  
Yarong Feng ◽  
Xing Chen ◽  
Liyi Jia ◽  
Xiruo Song ◽  
Hosam M. Mahmoud
Keyword(s):  
Polya Process ◽  
Pólya Process
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