There are at least two distinct notions of when a point process is stationary.
The most commonly utilized terminology is as follows: Intuitively, a point process 
defined on a subset 
of 
is said to be stationary if the number of points lying in
depends on the size of 
but not its location. On the real
line, this is expressed in terms of intervals: A
point process 
on 
is stationary if for all 
and for 
,
![Pr{N(t,t+x]=k}](/images/equations/StationaryPointProcess/NumberedEquation1.svg) |
depends on the length of 
but not on the location 
.
Stationary point processes of this kind were originally called simple stationary, though several authors call it crudely stationary instead. In light of the notion
of crude stationarity, a different definition of stationary may be stated in which
a point process 
is stationary whenever for every 
and for all bounded Borel subsets 
of 
, the joint distribution
of 
does not depend on 
. This distinction also gives rise to a related notion
known as interval stationarity.
Some authors use the alternative definition of an intensity function 
,
however, and conclude that a point process 
is stationary whenever 
is a constant function.
In this case, 
may also be called homogeneous or first order stationary (Pawlas 2008).
Other notions of stationarity exist for more general spaces as well; information on such spaces can be found in the work of, e.g., Daley and Vere-Jones (2007).
