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 $\mathrm{\Omega}$,

1.
$X(t)\in {\mathbb{Z}}^{+}\cup \{0\}$ for all $t$,
 2.

3.
$X(t)(\omega )$ is nondecreasing,

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 $$ and $X(t)(\omega )+1=X(s)(\omega )$.
Remark. For any $t$, the random variable^{} $X(t)$ is usually called the number of occurrences of some event by time $t$. Then, for $$, $X(t)X(s)$ is the number of occurrences in the halfopen interval $(s,t]$.
Title  counting process 

Canonical name  CountingProcess 
Date of creation  20130322 15:01:19 
Last modified on  20130322 15:01:19 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  5 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60G51 