independent increment

A stochastic process $\{X(t)\mid t\in T\}$ of real-valuedrandom variables $X(t)$, where $T$ is linearly ordered, is said haveindependent increments if for any $a,b,c,d\in T$ such that , $X(a)-X(b)$ and $X(c)-X(d)$ are independent randomvariables.

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 thecomparison of two positive quantities (for example, the numbers ofoccurrences of a certain event in some time intervals).