On the comparison of point processes

1985 ◽  
Vol 22 (2) ◽  
pp. 300-313 ◽  
Author(s):  
Y. L. Deng
Keyword(s):  
Point Processes ◽  
Counting Processes ◽  
Random Change Of Time ◽  
Random Change

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

On the comparison of point processes

1985 ◽  
Vol 22 (02) ◽  
pp. 300-313 ◽  
Author(s):  
Y. L. Deng
Keyword(s):  
Point Processes ◽  
Counting Processes ◽  
Random Change Of Time ◽  
Random Change

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4]. In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (04) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pk f of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tk f of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

On a class of multivariate counting processes

2016 ◽  
Vol 48 (2) ◽  
pp. 443-462 ◽  
Author(s):  
Ji Hwan Cha ◽  
Massimiliano Giorgio
Keyword(s):  
Point Processes ◽  
Dependence Structure ◽  
Counting Processes ◽  
New Class ◽  
Marginal Process ◽  
Polya Process ◽  
History Of ◽  
Pólya Process ◽  
Important Characterization

Abstract In this paper we define and study a new class of multivariate counting processes, named `multivariate generalized Pólya process'. Initially, we define and study the bivariate generalized Pólya process and briefly discuss its reliability application. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the process. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya process are obtained efficiently. The marginal processes of the multivariate generalized Pólya process are shown to be the univariate generalized Pólya processes studied in Cha (2014). Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya process is extended to the multivariate case. We define a new dependence concept for multivariate point processes and, based on it, we analyze the dependence structure of the multivariate generalized Pólya process.

Sticky Brownian motion as the limit of storage processes

1981 ◽  
Vol 18 (01) ◽  
pp. 216-226 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Brownian Motion ◽  
Infinitesimal Generator ◽  
Boundary Behavior ◽  
Compound Poisson Process ◽  
Storage Process ◽  
Reflected Brownian Motion ◽  
Jump Intensity ◽  
Random Change Of Time ◽  
Random Change ◽  
Storage Processes

The paper considers a modified storage processwith state space [0,∞). Away from the origin,Wbehaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls toIt is shown thatWcan be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motionW∗on [0,∞). ThusW∗is the natural diffusion approximation forW, and it is shown thatWconverges in distribution toW∗under appropriate conditions. The boundary behavior ofW∗is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (4) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

Sticky Brownian motion as the limit of storage processes

1981 ◽  
Vol 18 (1) ◽  
pp. 216-226 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Brownian Motion ◽  
Infinitesimal Generator ◽  
Boundary Behavior ◽  
Compound Poisson Process ◽  
Storage Process ◽  
Reflected Brownian Motion ◽  
Jump Intensity ◽  
Random Change Of Time ◽  
Random Change ◽  
Storage Processes

The paper considers a modified storage process with state space [0,∞). Away from the origin, W behaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls to It is shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motion W∗ on [0,∞). Thus W∗ is the natural diffusion approximation for W, and it is shown that W converges in distribution to W∗ under appropriate conditions. The boundary behavior of W∗ is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

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