## You are here

Homecounting process

## Primary tabs

# counting process

A stochastic process $\{X(t)\mid t\in\mathbb{R}^{{+}}\cup\{0\}\}$ is called a
*counting process* if, for each outcome $\omega$ in the sample space $\Omega$,

1. $X(t)\in\mathbb{Z}^{{+}}\cup\{0\}$ for all $t$,

2. 3. $X(t)(\omega)$ is non-decreasing,

4. $X(t)(\omega)$ is right continuous (continuous from the right), and

5. for any $t$, there is an $s\in\mathbb{R}$ such that $t<s$ and $X(t)(\omega)+1=X(s)(\omega)$.

Remark. For any $t$, the random variable $X(t)$ is usually called the number of occurrences of some event by time $t$. Then, for $s<t$, $X(t)-X(s)$ is the number of occurrences in the half-open interval $(s,t]$.

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

60G51*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections