(Stationary independent
increments) A process
is said to have "stationary independent increments" if
has a distribution depending only on
.

The notion of stationary independence is more restrictive the homogeneity of
increments because homogeneity only states the manner of time dependence and
leaves the possibility of
-dependence.

Definition

(Levy process) A Feller process with stationary
independent increments is called "Levy process". Equivalently,
is a "Levy process" if for any
the increment
is independent from
and has the same distribution as
.