independent increment

A stochastic process $\{X(t)\mid t\in T\}$ 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 , $X(a)-X(b)$ 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).

Title	independent increment
Canonical name	IndependentIncrement
Date of creation	2013-03-22 15:01:22
Last modified on	2013-03-22 15:01:22
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	4
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60G51