As an example, consider a Gaussian random walk with ,
where the finitely many
are jointly normally
distributed. Such a random walk can be constructed by assuming
identically, independently and normally distributed increments. By
the properties of the normal distribution, it follows that
is
-distributed for each
. If
and
is finite, it holds approximately for all
random walks for
large enough:
Random walks are processes with independent
increments. That means, the
increment of the process from time
to time
is
independent of the past values
up to time
.
In general, it holds for any
that the increment of the
process from time
to time
Processes with independent increments are also Markov-processes. In other words, the future evolution of the
process in time depends exclusively on
, and the value of
is independent of the past values
If
the increments
and the starting value
, and hence all
, can take a finite or countably infinite number of values,
then the Markov-property is formally expressed by: