 Counting process

A counting process is a stochastic process {N(t), t ≥ 0} that possesses the following properties:
 N(t) ≥ 0.
 N(t) is an integer.
 If s ≤ t then N(s) ≤ N(t).
If s < t, then N(t) − N(s) is the number of events occurred during the interval (s, t ]. Examples of counting processes include Poisson processes and Renewal processes.
Because of the third property, a counting process is increasing and hence a submartingale. Then by DoobMeyer, it can be written as
N(t)=M(t)+A(t)
with a martingale M(t) and a predictable increasing process A(t). The martingale M(t) is called the martingale associated with the counting process N(t) and the predictable process A(t) is called the cummulative intensity of the counting process N(t).
Counting processes deal with the number of various outcomes in a system over time. An example of a counting process is the number of occurrences of "heads" over some number of coin tosses.
If a process has the Markov property, it is said to be a Markov counting process.
References
 Ross, S.M. (1995) Stochastic Processes. Wiley. ISBN 9780471120629
 Higgins JJ, KellerMcNulty S (1995) Concepts in Probability and Stochastic Modeling. Wadsworth Publishing Company. ISBN 0534231365
Categories: Stochastic processes
 Statistical terminology
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