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independent increment (Definition)

A stochastic process $ \lbrace X(t)\mid t\in T\rbrace$ of real-valued random variables $ X(t)$, where $ T$ is linearly ordered, is said have independent increments if for any $ a,b,c,d\in T$ such that $ a<b<c<d$, $ X(a)-X(b)$ and $ X(c)-X(d)$ are independent random variables.

Remark. In case when $ X(t)$ is monotonically non-decreasing, as in the case of a counting process, it is customary to write $ X(b)-X(a)$ and $ X(d)-X(c)$ instead of the above to emphasize the comparison of two positive quantities (for example, the numbers of occurrences of a certain event in some time intervals).



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Cross-references: intervals, event, occurrences, numbers, positive, counting process, monotonically, independent, linearly ordered, random variables, stochastic process
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This is version 1 of independent increment, born on 2005-02-09.
Object id is 6731, canonical name is IndependentIncrement.
Accessed 3360 times total.

Classification:
AMS MSC: 60G51 (Probability theory and stochastic processes :: Stochastic processes :: Processes with independent increments)
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