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

Summing 

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Copyright information

© Chapman and Hall 1988

Authors and Affiliations

  • W. A. ThompsonJr
    • 1
  1. 1.University of MissouriColumbiaUSA

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