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counting process
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(Definition)
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A stochastic process $\lbrace X(t)\mid t\in \mathbb{R}^{+}\cup\lbrace 0 \rbrace \rbrace$ is called a counting process if, for each outcome $\omega$ in the sample space $\Omega$ ,
- $X(t)\in \mathbb{Z}^{+}\cup\lbrace 0 \rbrace$ for all $t$ ,
- $X(t)(\omega)$ is piecewise constant,
- $X(t)(\omega)$ is non-decreasing,
- $X(t)(\omega)$ is right continuous (continuous from the right), and
- 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 $(s,t]$ .
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"counting process" is owned by CWoo.
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Cross-references: half-open interval, event, occurrences, number, random variable, continuous from the right, continuous, right, piecewise, outcome, stochastic process
There are 2 references to this entry.
This is version 2 of counting process, born on 2005-02-09, modified 2005-02-15.
Object id is 6730, canonical name is CountingProcess.
Accessed 3855 times total.
Classification:
| AMS MSC: | 60G51 (Probability theory and stochastic processes :: Stochastic processes :: Processes with independent increments) |
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Pending Errata and Addenda
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