A point process 
on 
is said to be interval stationary if for every 
and for all integers 
,
the joint distribution of
 |
does not depend on 
, 
. Here, 
is an interval
for all 
.
As pointed out in a variety of literature (e.g., Daley and Vere-Jones 2002, pp 45-46), the notion of an interval stationary point process is intimately connected to (though fundamentally different from) the idea of a stationary point process in the Borel set sense of the term. Worth noting, too, is the difference between interval stationarity and other notions such as simple/crude stationarity.
Though it has been done, it is more difficult to extend to 
the notion of interval stationarity; doing so requires a
significant amount of additional machinery and reflects, overall, the significantly-increased
structural complexity of higher-dimensional Euclidean spaces (Daley and Vere-Jones
2007).
