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

$$ \Gamma = \mathbf R ^ {+} \setminus \cup ( x _ {i} ,x _ {i} + y _ {i} ) . $$

Here, the random cutouts $ ( x _ {i} ,x _ {i} + y _ {i} ) $ are associated with points $ ( x _ {i} ,y _ {i} ) $ that are randomly distributed in $ \mathbf R ^ {+} \times \mathbf R ^ {+} $ in such a way that their number in any given rectangle $ I \times J $ is a Poisson random variable (cf. Poisson process) with parameter $ \lambda ( I ) \mu ( J ) $, where $ \lambda $ is the Lebesgue measure and $ \mu $ is a given measure on $ \mathbf R ^ {+} $ that is locally bounded except at $ 0 $. The set of points $ ( x _ {i} ,y _ {i} ) $ is called a point Poisson process with intensity $ \lambda \otimes \mu $, and $ \Gamma $ can be viewed as the set of points in $ \mathbf R ^ {+} $ that are never in the shadow of the point Poisson process when light comes from the directions $ ( \cos \theta, \sin \theta ) $, $ {\pi / 2 } < \theta < { {3 \pi } / 4 } $.

The second random set is the closure of the range of a positive Lévy process with drift $ \gamma $ and Lévy measure $ \nu $. By definition, this process, $ L ( t ) $( $ = L ( t, \omega ) $), has independent and stationary increments, and

$$ {\mathsf E} e ^ {- uL ( t ) } = e ^ {- t \psi ( u ) } , $$

$$ \psi ( u ) = \gamma u + \int\limits _ { 0 } ^ \infty {( 1 - e ^ {- u z } ) } {\nu ( dz ) } . $$

The theorem asserts that for any given $ \mu $ with

$$ \tag{a1 } \int\limits _ { 0 } ^ { 1 } { { \mathop{\rm exp} } \left ( \int\limits _ { x } ^ { 1 } {\mu ( y, \infty ) } {dy } \right ) } {dx } < \infty, $$

one can explicitly define $ \gamma $ and $ \nu $ in such a way that

$$ \textrm{ law of } F = \textrm{ law of } {\overline{ {L ( \mathbf R ^ {+} ) }}\; } . $$

The drift $ \gamma $ vanishes precisely when

$$ \int\limits _ { 0 } ^ { 1 } {\mu ( z, \infty ) } {dz } = \infty. $$

For example, when $ \mu ( dy ) = { {a dy } / {y ^ {2} } } $, $ 0 < a < 1 $, then $ \gamma = 0 $ and $ \nu ( z, \infty ) = z ^ {- a } $( this is the case of a stable Lévy process of index $ 1 - a $).

When the integral in (a1) is infinite, a formal computation gives $ \gamma = 0 $ and $ \nu $ concentrated at $ + \infty $. This is the case when $ \Gamma $ 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 $ \Gamma \neq \emptyset $ almost surely [a3].

Now, given a compact subset $ K $ of $ \mathbf R ^ {+} $, the probabilities of the events $ \Gamma \cap K = \emptyset $ and $ {\overline{ {L ( \mathbf R ^ {+} ) }}\; } \cap K = \emptyset $ are the same; in other words: $ \Gamma \cap K = \emptyset $ almost surely if and only if $ K $ is a polar set for the Lévy process $ L ( t ) $. Since compact polar sets are precisely the compact sets of vanishing capacity with respect to a potential kernel (associated with $ \gamma $ and $ \nu $ and therefore with $ \mu $), the link between Poisson covering of $ \mathbf R ^ {+} $ 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