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Counting Process

A counting process is a stochastic process {N(t):t>=0} whose sample paths are nondecreasing, right-continuous, and nonnegative integer-valued, with N(0)=0. It models a stream of events by taking N(t) to be the number that have occurred by time t. Thus, for 0<=s<t, the increment

N(t)-N(s)

counts the events occurring in the interval (s,t].

If every jump has size one, the jump times of a counting process define a simple point process. Conversely, a locally finite simple temporal point process on [0,infty) with no point at 0 determines a counting process by counting its points in the interval (0,t] for t>=0 (Daley and Vere-Jones 2003).

See also

Point Process, Poisson Process, Simple Point Process, Stochastic Process, Temporal Point Process

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References

Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods, 2nd ed. New York: Springer, 2003.

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Counting Process

Cite this as:

Weisstein, Eric W. "Counting Process." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CountingProcess.html

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