scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (02) ◽  
pp. 473-475
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

scholarly journals The probability generating functional

1972 ◽  
Vol 14 (4) ◽  
pp. 448-466 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Generating Functional ◽  
The Real ◽  
The Subject

This paper is concerned with certain aspects of the theory and application of the probability generating functional (p.g.fl) of a point process on the real line. Interest in point processes has increased rapidly during the last decade and a number of different approaches to the subject have been expounded (see for example [6], [11], [15], [17], [20], [25], [27], [28]). It is hoped that the present development using the p.g.ff will calrify and unite some of these viewpoints and provide a useful tool for solution of particular problems.

Dependent thinning of point processes

1980 ◽  
Vol 17 (04) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

scholarly journals On a Model for the Storage of Files on a Hardware: Statistics at a Fixed Time and Asymptotic Regimes

2009 ◽  
Vol 2009 ◽  
pp. 1-24
Author(s):  
Vincent Bansaye
Keyword(s):  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Poisson Point Process ◽  
Fixed Time ◽  
The Real ◽  
Parking Problem ◽  
The Right

We consider a version in continuous time of the parking problem of Knuth. Files arrive following a Poisson point process and are stored on a hardware identified with the real line, in the closest free portions at the right of the arrival location. We specify the distribution of the space of unoccupied locations at a fixed time and give asymptotic regimes when the hardware is becoming full.

Dependent thinning of point processes

1980 ◽  
Vol 17 (4) ◽  
pp. 987-995 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Real Line ◽  
Point Processes ◽  
Compound Poisson Process ◽  
Second Order ◽  
Binary Variables ◽  
Compound Poisson ◽  
The Real ◽  
Markov Sequence

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (04) ◽  
pp. 734-743
Author(s):  
Toshio Mori
Keyword(s):  
Point Process ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Identification Problem ◽  
Original Process ◽  
The Real ◽  
Displacement Distribution ◽  
Stationary Point Process ◽  
Stationary Point Processes

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

scholarly journals On a Model for the Storage of Files on a Hardware. II. Evolution of a Typical Data Block

2007 ◽  
Vol 44 (4) ◽  
pp. 901-927 ◽  
Author(s):  
Vincent Bansaye
Keyword(s):  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Poisson Point Process ◽  
Data Block ◽  
The Real ◽  
Parking Problem ◽  
Typical Data ◽  
The Right ◽  
Arrival Point

We consider the generalized version in continuous time of the parking problem of Knuth introduced in Bansaye (2006). Files arrive following a Poisson point process and are stored on a hardware identified with the real line, at the right of their arrival point. Here we study the evolution of the endpoints of the data block straddling 0, which is empty at time 0 and is equal to R at a deterministic time.

scholarly journals On a Model for the Storage of Files on a Hardware. II. Evolution of a Typical Data Block

2007 ◽  
Vol 44 (04) ◽  
pp. 901-927
Author(s):  
Vincent Bansaye
Keyword(s):  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Poisson Point Process ◽  
Data Block ◽  
The Real ◽  
Parking Problem ◽  
Typical Data ◽  
The Right ◽  
Arrival Point

We consider the generalized version in continuous time of the parking problem of Knuth introduced in Bansaye (2006). Files arrive following a Poisson point process and are stored on a hardware identified with the real line, at the right of their arrival point. Here we study the evolution of the endpoints of the data block straddling 0, which is empty at time 0 and is equal to R at a deterministic time.

Ergodicity and identifiability for random translations of stationary point processes

1975 ◽  
Vol 12 (4) ◽  
pp. 734-743 ◽  
Author(s):  
Toshio Mori
Keyword(s):  
Point Process ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Identification Problem ◽  
Original Process ◽  
The Real ◽  
Displacement Distribution ◽  
Stationary Point Process ◽  
Stationary Point Processes

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.

Sign in / Sign up

Export Citation Format

Share Document