Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Counting process

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated May 10, 2025

A counting process is a stochastic process \{N(t),t\geq 0\} with values that are non-negative, integer, and non-decreasing:

  1. N(t)\geq 0.
  2. N(t) is an integer.
  3. If s\leq t then N(s)\leq 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.

Counting processes deal with the number of occurrences of something over time. An example of a counting process is the number of job arrivals to a queue over time.

If a process has the Markov property, it is said to be a Markov counting process.

Watch videos about Counting processExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Counting process”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Intensity of counting processes

The intensity λ {\displaystyle \lambda } of a counting process is a measure of the rate of change of its predictable part. If a stochastic process { N , t ≥ 0 } {\displaystyle \{N(t),t\geq 0\}} is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is N ( t ) = M ( t ) + Λ ( t ) {\displaystyle N(t)=M(t)+\Lambda (t)} where M ( t ) {\displaystyle M(t)} is a martingale and Λ ( t ) {\displaystyle \Lambda (t)} is a predictable increasing process.

Poisson point process

In probability theory, statistics and related fields, a Poisson point process is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution.