Intensity Function

There are at least two distinct notions of an intensity function related to the theory of point processes.

In some literature, the intensity  of a point process  is defined to be the quantity

(1)

provided it exists. Here,  denotes probability. In particular, it makes sense to talk about point processes having infinite intensity, though when finite,  allows  to be rewritten so that

(2)

as  where here,  denotes little-O notation (Daley and Vere-Jones 2007).

Other authors define the function  to be an intensity function of a point process  provided that  is a density of the intensity measure  associated to  relative to Lebesgue measure, i.e.,if for all Borel sets  in ,

(3)

where  denotes Lebesgue measure (Pawlas 2008).

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.

Pawlas, Z. "Spatial Modeling and Spatial Statistics." Course Notes. Autumn 2008. https://www.math.ku.dk/~pawlas/rumlig.pdf.