The Inverse Square Root Law

A spatial Poisson process is defined in a way similar to the manner by which a temporal process was treated in the preceding vignette. Suppose events take place in the plane at random and that if S is any subset of the plane, then N(S) counts the number of events that occurs within S. If S and S′ are disjoint subsets then we require that N(S) and N(S′) be independent random variables (independent increments) and that N(S) depend only on the area of S and not on its position or shape (stationary increments).The spatial counting process is called Poisson if