counting process

A stochastic processMathworldPlanetmath {X(t)t+{0}} is called a counting processMathworldPlanetmath if, for each outcome ω in the sample space Ω,

  1. 1.

    X(t)+{0} for all t,

  2. 2.

    X(t)(ω) is piecewise constant,

  3. 3.

    X(t)(ω) is non-decreasing,

  4. 4.

    X(t)(ω) is right continuousMathworldPlanetmathPlanetmath (continuous from the right), and

  5. 5.

    for any t, there is an s such that t<s and X(t)(ω)+1=X(s)(ω).

Remark. For any t, the random variableMathworldPlanetmath 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].

Title counting process
Canonical name CountingProcess
Date of creation 2013-03-22 15:01:19
Last modified on 2013-03-22 15:01:19
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 60G51