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''Skorokhod representation theorem''
 
''Skorokhod representation theorem''
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101901.png" /> is a sequence of probability measures on a complete and separable [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101902.png" /> that converges weakly (cf. [[Weak topology|Weak topology]]) to a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101903.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101904.png" /> for any continuous and bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101905.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101906.png" />). Then there exists a [[Probability space|probability space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101908.png" />-valued random elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s1101909.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019010.png" /> with distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019012.png" />, respectively, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019013.png" /> converges <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019014.png" />-almost surely to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019015.png" /> (cf. [[Convergence, almost-certain|Convergence, almost-certain]]).
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Suppose that $  \{ P _ {n} \} _ {n \geq  1 }  $
 +
is a sequence of probability measures on a complete and separable [[Metric space|metric space]] $  ( S, {\mathcal S} ) $
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that converges weakly (cf. [[Weak topology|Weak topology]]) to a [[Probability measure|probability measure]] $  P $(
 +
that is, $  {\lim\limits } _ {n} \int _ {S} f  {dP _ {n} } = \int _ {S} f  {dP } $
 +
for any continuous and bounded function $  f $
 +
on $  S $).  
 +
Then there exists a [[Probability space|probability space]] $  ( \Omega, {\mathcal F}, {\mathsf P} ) $
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and $  S $-
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valued random elements $  \{ X _ {n} \} $,  
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$  X $
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with distributions $  \{ P _ {n} \} $
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and $  P $,  
 +
respectively, such that $  X _ {n} $
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converges $  {\mathsf P} $-
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almost surely to $  X $(
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cf. [[Convergence, almost-certain|Convergence, almost-certain]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019016.png" />, the proof of this result reduces to taking for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019017.png" /> the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019018.png" /> with [[Lebesgue measure|Lebesgue measure]] and letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019020.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019021.png" />.
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If $  S = \mathbf R $,  
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the proof of this result reduces to taking for $  \Omega $
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the unit interval $  ( 0,1 ) $
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with [[Lebesgue measure|Lebesgue measure]] and letting $  X _ {n} ( y ) = \inf  \{ z : {P _ {n} ( - \infty,z ] \geq  y } \} $,  
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and $  X ( y ) = \inf  \{ z : {P ( - \infty,z ] \geq  y } \} $,  
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for $  y \in ( 0,1 ) $.
  
In [[#References|[a1]]] the theorem has been extended to general separable metric spaces, while in [[#References|[a4]]] the result is proved for an arbitrary metric space, assuming that the limit probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110190/s11019022.png" /> is concentrated on a separable set. Extensions of this theorem to non-metrizable topological spaces are discussed in [[#References|[a2]]].
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In [[#References|[a1]]] the theorem has been extended to general separable metric spaces, while in [[#References|[a4]]] the result is proved for an arbitrary metric space, assuming that the limit probability $  P $
 +
is concentrated on a separable set. Extensions of this theorem to non-metrizable topological spaces are discussed in [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Dudley,  "Distance of probability measures and random variables"  ''Ann. Math. Stat.'' , '''39'''  (1968)  pp. 1563–1572</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schief,  "Almost surely convergent random variables with given laws"  ''Probab. Th. Rel. Fields'' , '''81'''  (1989)  pp. 559–567</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.V. Skorokhod,  "Limit theorems for stochastic processes"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 261–290</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.J. Wichura,  "On the construction of almost uniformly convergent random variables with given weakly convergent image laws"  ''Ann. Math. Stat.'' , '''41'''  (1970)  pp. 284–291</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.M. Dudley,  "Distance of probability measures and random variables"  ''Ann. Math. Stat.'' , '''39'''  (1968)  pp. 1563–1572</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schief,  "Almost surely convergent random variables with given laws"  ''Probab. Th. Rel. Fields'' , '''81'''  (1989)  pp. 559–567</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.V. Skorokhod,  "Limit theorems for stochastic processes"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 261–290</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.J. Wichura,  "On the construction of almost uniformly convergent random variables with given weakly convergent image laws"  ''Ann. Math. Stat.'' , '''41'''  (1970)  pp. 284–291</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


Skorokhod representation theorem

Suppose that $ \{ P _ {n} \} _ {n \geq 1 } $ is a sequence of probability measures on a complete and separable metric space $ ( S, {\mathcal S} ) $ that converges weakly (cf. Weak topology) to a probability measure $ P $( that is, $ {\lim\limits } _ {n} \int _ {S} f {dP _ {n} } = \int _ {S} f {dP } $ for any continuous and bounded function $ f $ on $ S $). Then there exists a probability space $ ( \Omega, {\mathcal F}, {\mathsf P} ) $ and $ S $- valued random elements $ \{ X _ {n} \} $, $ X $ with distributions $ \{ P _ {n} \} $ and $ P $, respectively, such that $ X _ {n} $ converges $ {\mathsf P} $- almost surely to $ X $( cf. Convergence, almost-certain).

If $ S = \mathbf R $, the proof of this result reduces to taking for $ \Omega $ the unit interval $ ( 0,1 ) $ with Lebesgue measure and letting $ X _ {n} ( y ) = \inf \{ z : {P _ {n} ( - \infty,z ] \geq y } \} $, and $ X ( y ) = \inf \{ z : {P ( - \infty,z ] \geq y } \} $, for $ y \in ( 0,1 ) $.

In [a1] the theorem has been extended to general separable metric spaces, while in [a4] the result is proved for an arbitrary metric space, assuming that the limit probability $ P $ is concentrated on a separable set. Extensions of this theorem to non-metrizable topological spaces are discussed in [a2].

References

[a1] R.M. Dudley, "Distance of probability measures and random variables" Ann. Math. Stat. , 39 (1968) pp. 1563–1572
[a2] A. Schief, "Almost surely convergent random variables with given laws" Probab. Th. Rel. Fields , 81 (1989) pp. 559–567
[a3] A.V. Skorokhod, "Limit theorems for stochastic processes" Th. Probab. Appl. , 1 (1956) pp. 261–290
[a4] M.J. Wichura, "On the construction of almost uniformly convergent random variables with given weakly convergent image laws" Ann. Math. Stat. , 41 (1970) pp. 284–291
How to Cite This Entry:
Skorokhod theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skorokhod_theorem&oldid=48732
This article was adapted from an original article by D. Nualart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article