Let P1 be a Poisson process of intensity λ1 on the positive x-axis, P2 a Poisson process of intensity λ2 on the positive y-axis, and P a Poisson process of intensity λ1λ2 in the interior of IR +, where P1, P2 and P are independent. Then the extended Hammersley process Lλ1(·, t) with sources and sinks given by P1 and P2, respectively, is distributed as a Poisson point process with intensity λ1 f...

1. Examples 1.1. Poisson Process 1.2. Markov Chain 1.3. Martingale 1.4. Brownian Motion 2. Definition of the Stochastic Process 3. Poisson Process 3.1. The Distribution 3.2. A Construction of the Poisson Process 3.3. Generalizations 4. Brownian Motion 4.1. Random Walk 4.2. Distributions 4.3. Path Properties 4.4. Local Time 4.5. Fourier Series Representation Glossary Bibliography Boigraphical Sk...

The Poisson process is a stochastic counting process that arises naturally in a large variety of daily-life situations. We present a few definitions of the Poisson process and discuss several properties as well as relations to some well-known probability distributions. We further briefly discuss the compound Poisson process.

We present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172– 182] for sums of uniformly null point processes that have Poisson limits. Included is an application in which a particle-survivor point process converges to a Poisson process. This result she...

Change point estimation in the area of statistical process control has received considerable attentions in the recent decades because it helps process engineer to identify and remove assignable causes as quickly as possible. On the other hand, improving in measurement systems and data storage, lead to taking observations very close to each other in time and as a result increasing autocorrelatio...

Abstract: This paper explores large sample properties of the two parameter (α, θ) Poisson-Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two parameter Poisson Dirichlet posterior process. We also establish the we...

This paper analyzes statistical properties of the Poisson line Cox point process useful in the modeling of vehicular networks. The point process is created by a two-stage construction: a Poisson line process to model road infrastructure and independent Poisson point processes, conditionally on the Poisson lines, to model vehicles on the roads. We derive basic properties of the point process, in...

  Multistage process monitoring has recently attracted notable attention in that the statistical relationships between quality variables are taken into account. Here, we dealt with the problem of monitoring and diagnosing a two-stage production process with attribute characteristics in which the outgoing quality variable is impacted by the incoming quality variable from the first process stage....