Interval Stationary Point Process

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).

REFERENCES:

Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods, 2nd ed. New York: Springer, 2003.

Daley, D. J. and Vere-Jones, D. An Introduction to the Theory of Point Processes Volume II: General Theory and Structure, 2nd ed. New York: Springer, 2007.