# Markov point processes

• W. A. ThompsonJr

## Abstract

A generalization of the independent increments property is to allow the distribution of the number of arrivals in an interval to depend on the number of arrivals accumulated at the beginning of the interval. That is, for all t1 < … < tn and all n:
$$P\left[ {N\left( {{t_{n - 1}},{t_n}} \right)} \right] = k|N\left( {{t_{n - 1}}} \right) = {k_{n - 1}}, \ldots ,N\left( {{t_1}} \right) = {k_1}] = P\left[ {N\left( {{t_{n - 1}},{t_n}} \right)} \right] = k|N\left( {{t_{n - 1}}} \right) = {k_{n - 1}}]$$
or equivalently
$$P\left[ {N\left( {{t_n}} \right) = {k_n}|N\left( {{t_{n - 1}}} \right) = {k_{n - 1}}, \ldots ,N\left( {t{}_1} \right) = {k_1}} \right] = P\left[ {N\left( {{t_n}} \right) = {k_n}|N\left( {{t_{n - 1}}} \right) = k{}_{n - 1}} \right]$$
This last is called the Markov property; if it is satisfied we have a Markov point process.

## Keywords

Markov Process POISSON Process Markov Property Negative Binomial Distribution Conditional Intensity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.