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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]]).

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|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  $  \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 [[#References|[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|Billard method]]).

====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>