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A theorem asserting that two particular random sets obtained in quite different ways have the same distribution law [[#References|[a1]]]. A first version of this theorem was obtained by B. Mandelbrot in 1972 [[#References|[a2]]]. It is a key fact for understanding the link between random coverings and [[Potential theory|potential theory]] (see also [[Dvoretzky problem|Dvoretzky problem]]; [[Billard method|Billard method]]).
 
A theorem asserting that two particular random sets obtained in quite different ways have the same distribution law [[#References|[a1]]]. A first version of this theorem was obtained by B. Mandelbrot in 1972 [[#References|[a2]]]. It is a key fact for understanding the link between random coverings and [[Potential theory|potential theory]] (see also [[Dvoretzky problem|Dvoretzky problem]]; [[Billard method|Billard method]]).
  
 
The first random set is defined as
 
The first random set is defined as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100901.png" /></td> </tr></table>
+
$$
 +
\Gamma = \mathbf R  ^ {+} \setminus  \cup ( x _ {i} ,x _ {i} + y _ {i} ) .
 +
$$
  
Here, the random cutouts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100902.png" /> are associated with points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100903.png" /> that are randomly distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100904.png" /> in such a way that their number in any given rectangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100905.png" /> is a Poisson random variable (cf. [[Poisson process|Poisson process]]) with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100906.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100907.png" /> is the [[Lebesgue measure|Lebesgue measure]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100908.png" /> is a given measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f1100909.png" /> that is locally bounded except at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009010.png" />. The set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009011.png" /> is called a point Poisson process with intensity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009013.png" /> can be viewed as the set of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009014.png" /> that are never in the shadow of the point Poisson process when light comes from the directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009016.png" />.
+
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|Poisson process]]) with parameter $  \lambda ( I ) \mu ( J ) $,  
 +
where $  \lambda $
 +
is the [[Lebesgue measure|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009017.png" /> and Lévy measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009018.png" />. By definition, this process, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009019.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009020.png" />), has independent and stationary increments, and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009021.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} e ^ {- uL ( t ) } = e ^ {- t \psi ( u ) } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009022.png" /></td> </tr></table>
+
$$
 +
\psi ( u ) = \gamma u + \int\limits _ { 0 } ^  \infty  {( 1 - e ^ {- u z } ) }  {\nu ( dz ) } .
 +
$$
  
The theorem asserts that for any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009023.png" /> with
+
The theorem asserts that for any given $  \mu $
 +
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\int\limits _ { 0 } ^ { 1 }  { { \mathop{\rm exp} } \left ( \int\limits _ { x } ^ { 1 }  {\mu ( y, \infty ) }  {dy } \right ) }  {dx } < \infty,
 +
$$
  
one can explicitly define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009026.png" /> in such a way that
+
one can explicitly define $  \gamma $
 +
and $  \nu $
 +
in such a way that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009027.png" /></td> </tr></table>
+
$$
 +
\textrm{ law  of  }  F = \textrm{ law  of  }  {\overline{ {L ( \mathbf R  ^ {+} ) }}\; } .
 +
$$
  
The drift <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009028.png" /> vanishes precisely when
+
The drift $  \gamma $
 +
vanishes precisely when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009029.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { 1 }  {\mu ( z, \infty ) }  {dz } = \infty.
 +
$$
  
For example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009033.png" /> (this is the case of a stable Lévy process of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009034.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009036.png" /> concentrated at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009037.png" />. This is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009038.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009039.png" /> almost surely [[#References|[a3]]].
+
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 [[#References|[a3]]].
  
Now, given a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009041.png" />, the probabilities of the events <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009043.png" /> are the same; in other words: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009044.png" /> almost surely if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009045.png" /> is a polar set for the Lévy process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009046.png" />. Since compact polar sets are precisely the compact sets of vanishing capacity with respect to a potential kernel (associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009048.png" /> and therefore with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009049.png" />), the link between Poisson covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110090/f11009050.png" /> and potential theory is manifest (see also [[Billard method|Billard method]]).
+
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|Billard method]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.B. Mandelbrot,  "Renewal sets and random cutouts"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''22'''  (1972)  pp. 145–157</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.A. Shepp,  "Covering the line by random intervals"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''23'''  (1972)  pp. 163–170</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.B. Mandelbrot,  "Renewal sets and random cutouts"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''22'''  (1972)  pp. 145–157</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.A. Shepp,  "Covering the line by random intervals"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''23'''  (1972)  pp. 163–170</TD></TR></table>

Latest revision as of 19:39, 5 June 2020


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
How to Cite This Entry:
Fitzsimmons-Fristedt-Shepp theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fitzsimmons-Fristedt-Shepp_theorem&oldid=18589
This article was adapted from an original article by J.-P. Kahane (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article