Fitzsimmons-Fristedt-Shepp theorem

A theorem asserting that two particular random sets obtained in quite different ways have the same distribution law [a1]. A first version of this theorem was obtained by B. Mandelbrot in 1972 [a2]. It is a key fact for understanding the link between random coverings and potential theory (see also Dvoretzky problem; Billard method).

The first random set is defined as

Here, the random cutouts are associated with points that are randomly distributed in in such a way that their number in any given rectangle is a Poisson random variable (cf. Poisson process) with parameter , where is the Lebesgue measure and is a given measure on that is locally bounded except at . The set of points is called a point Poisson process with intensity , and can be viewed as the set of points in that are never in the shadow of the point Poisson process when light comes from the directions , .

The second random set is the closure of the range of a positive Lévy process with drift and Lévy measure . By definition, this process, (), has independent and stationary increments, and

The theorem asserts that for any given with

(a1)

one can explicitly define and in such a way that

The drift vanishes precisely when

For example, when , , then and (this is the case of a stable Lévy process of index ).

When the integral in (a1) is infinite, a formal computation gives and concentrated at . This is the case when is empty. Therefore, the Fitzsimmons–Fristedt–Shepp theorem is an extension of Shepp's theorem, which states that (a1) is a necessary and sufficient condition for almost surely [a3].

Now, given a compact subset of , the probabilities of the events and are the same; in other words: almost surely if and only if is a polar set for the Lévy process . Since compact polar sets are precisely the compact sets of vanishing capacity with respect to a potential kernel (associated with and and therefore with ), the link between Poisson covering of and potential theory is manifest (see also Billard method).

References

[a1]	P.J. Fitzsimons, B. Fristedt, L.R. Shepp, "The set of real numbers left uncovered by random covering intervals" Z. Wahrscheinlichkeitsth. verw. Gebiete , 70 (1985) pp. 175–189
[a2]	B.B. Mandelbrot, "Renewal sets and random cutouts" Z. Wahrscheinlichkeitsth. verw. Gebiete , 22 (1972) pp. 145–157
[a3]	L.A. Shepp, "Covering the line by random intervals" Z. Wahrscheinlichkeitsth. verw. Gebiete , 23 (1972) pp. 163–170