Difference between revisions of "Fitzsimmons-Fristedt-Shepp theorem"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Fitzsimmons–Fristedt–Shepp theorem to Fitzsimmons-Fristedt-Shepp theorem: ascii title) |
(No difference)
|
Revision as of 18:52, 24 March 2012
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 |
Fitzsimmons-Fristedt-Shepp theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fitzsimmons-Fristedt-Shepp_theorem&oldid=22429