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A metric in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583201.png" /> of finite Borel measures (cf. [[Borel measure|Borel measure]]) on a [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583202.png" />, defined by
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<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/l/l058/l058320/l0583203.png" /></td> </tr></table>
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{{TEX|done}}
  
<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/l/l058/l058320/l0583204.png" /></td> </tr></table>
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A metric in the space  $  \mathfrak M $
 +
of finite Borel measures (cf. [[Borel measure|Borel measure]]) on a [[Metric space|metric space]]  $  ( U , d ) $,
 +
defined by
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583205.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583206.png" />-algebra of Borel sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583207.png" /> and
+
$$
 +
\pi ( P , Q ) =
 +
$$
  
<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/l/l058/l058320/l0583208.png" /></td> </tr></table>
+
$$
 +
= \
 +
\inf
 +
\{  \epsilon  : {P ( A) \leq  Q ( A  ^  \epsilon  ) +
 +
\epsilon , Q ( A) \leq  P ( A  ^  \epsilon  ) + \epsilon
 +
\textrm{ for  all  }  A \subset  \mathfrak B } \} ,
 +
$$
  
The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [[#References|[1]]] as a generalization of the [[Lévy metric|Lévy metric]]. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l0583209.png" /> changes if in its definition one omits one of the two inequalities and replaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832010.png" /> by the system of all open or closed sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832011.png" /> (see [[#References|[2]]]).
+
where  $  \mathfrak B $
 +
is the  $  \sigma $-
 +
algebra of Borel sets of  $  ( U , d ) $
 +
and
 +
 
 +
$$
 +
A  ^  \epsilon  =  \{ {x } : {d ( x , y ) < \epsilon , y \in A } \}
 +
.
 +
$$
 +
 
 +
The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [[#References|[1]]] as a generalization of the [[Lévy metric|Lévy metric]]. The quantity $  \pi $
 +
changes if in its definition one omits one of the two inequalities and replaces $  \mathfrak B $
 +
by the system of all open or closed sets of $  \mathfrak B $(
 +
see [[#References|[2]]]).
  
 
==Most important properties of the Lévy–Prokhorov metric.==
 
==Most important properties of the Lévy–Prokhorov metric.==
  
 +
1) The metric space  $  ( \mathfrak M , \pi ) $
 +
is separable if and only if  $  ( U , d ) $
 +
is separable (cf. [[Separable space|Separable space]]).
  
1) The metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832012.png" /> is separable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832013.png" /> is separable (cf. [[Separable space|Separable space]]).
+
2) The space $  ( U , d ) $
 +
is complete if the space  $  ( \mathfrak M , \pi ) $
 +
is complete (cf. [[Complete space|Complete space]]). The converse is true if the measures of  $  \mathfrak M $
 +
have separable supports.
  
2) The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832014.png" /> is complete if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832015.png" /> is complete (cf. [[Complete space|Complete space]]). The converse is true if the measures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832016.png" /> have separable supports.
+
3) In the space $  \mathfrak M $
 +
of probability measures the Lévy–Prokhorov metric has properties analogous to those of the Lévy metric. Namely, the regularity property 3) (cf. [[Lévy metric|Lévy metric]]) and its corollaries, properties 4) and 5), property 6) (in the case  $  U = \mathbf R  ^ {1} $),
 +
property 7) in part (namely,  $  \pi \leq  \mathop{\rm var} $),
 +
and also an analogue of property 8) if  $  ( U , d ) $
 +
is a linear normed space: If  $  P _ {a , \sigma }  ( A) = P ( \sigma A + a ) $,
 +
where  $  \sigma > 0 $,
 +
$  a \in U $,
 +
then for any  $  P , Q \in \mathfrak M $,
  
3) In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832017.png" /> of probability measures the Lévy–Prokhorov metric has properties analogous to those of the Lévy metric. Namely, the regularity property 3) (cf. [[Lévy metric|Lévy metric]]) and its corollaries, properties 4) and 5), property 6) (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832018.png" />), property 7) in part (namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832019.png" />), and also an analogue of property 8) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832020.png" /> is a linear normed space: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832023.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832024.png" />,
+
$$
 +
\pi ( \sigma P , \sigma Q ) \leq  \sigma \pi ( P _ {a , \sigma }  ,\
 +
Q _ {a , \sigma }  ) ,
 +
$$
  
<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/l/l058/l058320/l05832025.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\sigma \rightarrow 0 }  \pi ( P _ {a , \sigma }
 +
, Q _ {a , \sigma }  )  =   \mathop{\rm var} ( P , Q ) .
 +
$$
  
<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/l/l058/l058320/l05832026.png" /></td> </tr></table>
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4) In the case $  U = \mathbf R  ^ {k} $
 
+
the Lévy–Prokhorov metric in $  \mathfrak M $
4) In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832027.png" /> the Lévy–Prokhorov metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832028.png" /> can be estimated by means of the characteristic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832030.png" /> corresponding to the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832032.png" /> (see [[#References|[3]]], [[#References|[4]]]).
+
can be estimated by means of the characteristic functions $  f $
 +
and $  g $
 +
corresponding to the measures $  P $
 +
and $  Q $(
 +
see [[#References|[3]]], [[#References|[4]]]).
  
 
5) The Lévy–Prokhorov metric is a minimal metric with respect to the probability distance
 
5) The Lévy–Prokhorov metric is a minimal metric with respect to the probability distance
  
<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/l/l058/l058320/l05832033.png" /></td> </tr></table>
+
$$
 +
\kappa ( X , Y )  = \inf
 +
\{  \epsilon  : {P \{ d ( X , Y ) > \epsilon \}
 +
< \epsilon } \} ,
 +
$$
  
that is, for any random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832035.png" /> with fixed marginal distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832036.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832037.png" /> over all joint distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058320/l05832038.png" />.
+
that is, for any random variables $  X $,  
 +
$  Y $
 +
with fixed marginal distributions $  P _ {X} , P _ {Y} \in \mathfrak M $
 +
one has $  \pi ( P _ {X} , P _ {Y} ) = \inf  \kappa ( X , Y ) $
 +
over all joint distributions $  P _ {XY} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Prokhorov,  "Convergence of random processes and limit theorems in probability theory"  ''Theory Probab. Appl.'' , '''1'''  (1956)  pp. 157–214  ''Teor. Veroyatnost. i Primenen.'' , '''1''' :  2  (1956)  pp. 177–238</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.M. Dudley,  "Distances of probability measures and random variables"  ''Ann. Math. Stat.'' , '''39'''  (1968)  pp. 1563–1572</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Yurinskii,  "A smoothing inequality for estimates of the Lévy–Prokhorov distance"  ''Theory Probab. Appl.'' , '''20'''  (1975)  pp. 1–10  ''Teor. Veroyatnost. i Primenen.'' , '''20''' :  1  (1975)  pp. 3–12</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Abramov,  "Estimates for the Lévy–Prokhorov distance"  ''Theory Probab. Appl.'' , '''21'''  (1976)  pp. 396–400  ''Teor. Veroyatnost. i Primenen.'' , '''21''' :  2  (1976)  pp. 406–410</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Strassen,  "The existence of probability measures with given marginals"  ''Ann. Math. Stat.'' , '''36''' :  2  (1965)  pp. 423–439</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Prokhorov,  "Convergence of random processes and limit theorems in probability theory"  ''Theory Probab. Appl.'' , '''1'''  (1956)  pp. 157–214  ''Teor. Veroyatnost. i Primenen.'' , '''1''' :  2  (1956)  pp. 177–238</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.M. Dudley,  "Distances of probability measures and random variables"  ''Ann. Math. Stat.'' , '''39'''  (1968)  pp. 1563–1572</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Yurinskii,  "A smoothing inequality for estimates of the Lévy–Prokhorov distance"  ''Theory Probab. Appl.'' , '''20'''  (1975)  pp. 1–10  ''Teor. Veroyatnost. i Primenen.'' , '''20''' :  1  (1975)  pp. 3–12</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Abramov,  "Estimates for the Lévy–Prokhorov distance"  ''Theory Probab. Appl.'' , '''21'''  (1976)  pp. 396–400  ''Teor. Veroyatnost. i Primenen.'' , '''21''' :  2  (1976)  pp. 406–410</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Strassen,  "The existence of probability measures with given marginals"  ''Ann. Math. Stat.'' , '''36''' :  2  (1965)  pp. 423–439</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
See also [[Distributions, convergence of|Distributions, convergence of]]; [[Convergence of measures|Convergence of measures]]; [[Weak convergence of probability measures|Weak convergence of probability measures]].
 
See also [[Distributions, convergence of|Distributions, convergence of]]; [[Convergence of measures|Convergence of measures]]; [[Weak convergence of probability measures|Weak convergence of probability measures]].

Latest revision as of 04:11, 6 June 2020


A metric in the space $ \mathfrak M $ of finite Borel measures (cf. Borel measure) on a metric space $ ( U , d ) $, defined by

$$ \pi ( P , Q ) = $$

$$ = \ \inf \{ \epsilon : {P ( A) \leq Q ( A ^ \epsilon ) + \epsilon , Q ( A) \leq P ( A ^ \epsilon ) + \epsilon \textrm{ for all } A \subset \mathfrak B } \} , $$

where $ \mathfrak B $ is the $ \sigma $- algebra of Borel sets of $ ( U , d ) $ and

$$ A ^ \epsilon = \{ {x } : {d ( x , y ) < \epsilon , y \in A } \} . $$

The Lévy–Prokhorov metric was introduced by Yu.V. Prokhorov [1] as a generalization of the Lévy metric. The quantity $ \pi $ changes if in its definition one omits one of the two inequalities and replaces $ \mathfrak B $ by the system of all open or closed sets of $ \mathfrak B $( see [2]).

Most important properties of the Lévy–Prokhorov metric.

1) The metric space $ ( \mathfrak M , \pi ) $ is separable if and only if $ ( U , d ) $ is separable (cf. Separable space).

2) The space $ ( U , d ) $ is complete if the space $ ( \mathfrak M , \pi ) $ is complete (cf. Complete space). The converse is true if the measures of $ \mathfrak M $ have separable supports.

3) In the space $ \mathfrak M $ of probability measures the Lévy–Prokhorov metric has properties analogous to those of the Lévy metric. Namely, the regularity property 3) (cf. Lévy metric) and its corollaries, properties 4) and 5), property 6) (in the case $ U = \mathbf R ^ {1} $), property 7) in part (namely, $ \pi \leq \mathop{\rm var} $), and also an analogue of property 8) if $ ( U , d ) $ is a linear normed space: If $ P _ {a , \sigma } ( A) = P ( \sigma A + a ) $, where $ \sigma > 0 $, $ a \in U $, then for any $ P , Q \in \mathfrak M $,

$$ \pi ( \sigma P , \sigma Q ) \leq \sigma \pi ( P _ {a , \sigma } ,\ Q _ {a , \sigma } ) , $$

$$ \lim\limits _ {\sigma \rightarrow 0 } \pi ( P _ {a , \sigma } , Q _ {a , \sigma } ) = \mathop{\rm var} ( P , Q ) . $$

4) In the case $ U = \mathbf R ^ {k} $ the Lévy–Prokhorov metric in $ \mathfrak M $ can be estimated by means of the characteristic functions $ f $ and $ g $ corresponding to the measures $ P $ and $ Q $( see [3], [4]).

5) The Lévy–Prokhorov metric is a minimal metric with respect to the probability distance

$$ \kappa ( X , Y ) = \inf \{ \epsilon : {P \{ d ( X , Y ) > \epsilon \} < \epsilon } \} , $$

that is, for any random variables $ X $, $ Y $ with fixed marginal distributions $ P _ {X} , P _ {Y} \in \mathfrak M $ one has $ \pi ( P _ {X} , P _ {Y} ) = \inf \kappa ( X , Y ) $ over all joint distributions $ P _ {XY} $.

References

[1] Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Theory Probab. Appl. , 1 (1956) pp. 157–214 Teor. Veroyatnost. i Primenen. , 1 : 2 (1956) pp. 177–238
[2] R.M. Dudley, "Distances of probability measures and random variables" Ann. Math. Stat. , 39 (1968) pp. 1563–1572
[3] V.V. Yurinskii, "A smoothing inequality for estimates of the Lévy–Prokhorov distance" Theory Probab. Appl. , 20 (1975) pp. 1–10 Teor. Veroyatnost. i Primenen. , 20 : 1 (1975) pp. 3–12
[4] V.A. Abramov, "Estimates for the Lévy–Prokhorov distance" Theory Probab. Appl. , 21 (1976) pp. 396–400 Teor. Veroyatnost. i Primenen. , 21 : 2 (1976) pp. 406–410
[5] V. Strassen, "The existence of probability measures with given marginals" Ann. Math. Stat. , 36 : 2 (1965) pp. 423–439
[6] P. Billingsley, "Convergence of probability measures" , Wiley (1968)

Comments

See also Distributions, convergence of; Convergence of measures; Weak convergence of probability measures.

How to Cite This Entry:
Lévy-Prokhorov metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy-Prokhorov_metric&oldid=47734
This article was adapted from an original article by V.M. Zolotarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article