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A [[Topological structure (topology)|topological structure (topology)]] on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102001.png" /> of right-continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102002.png" /> having limits to the left at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102003.png" />, introduced by A.V. Skorokhod [[#References|[a4]]] as an alternative to the topology of [[Uniform convergence|uniform convergence]] in order to study the [[Convergence in distribution|convergence in distribution]] of stochastic processes with jumps.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102004.png" /> be the class of strictly increasing, continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102005.png" /> onto itself. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102006.png" /> one defines
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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/s/s110/s110200/s1102007.png" /></td> </tr></table>
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A [[Topological structure (topology)|topological structure (topology)]] on the space  $  D [ 0,1 ] $
 +
of right-continuous functions on  $  [ 0,1 ] $
 +
having limits to the left at each  $  t \in ( 0,1 ] $,
 +
introduced by A.V. Skorokhod [[#References|[a4]]] as an alternative to the topology of [[Uniform convergence|uniform convergence]] in order to study the [[Convergence in distribution|convergence in distribution]] of stochastic processes with jumps.
  
The following distance, introduced by P. Billingsley [[#References|[a1]]], induces the Skorokhod topology and makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s1102008.png" /> a complete separable [[Metric space|metric space]]:
+
Let  $  \Lambda $
 +
be the class of strictly increasing, continuous mappings of  $  [ 0,1 ] $
 +
onto itself. For  $  \lambda \in \Lambda $
 +
one defines
  
<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/s/s110/s110200/s1102009.png" /></td> </tr></table>
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$$
 +
\left \| \lambda \right \| = \sup  _ {s \neq t } \left | { { \mathop{\rm log} } {
 +
\frac{\lambda ( t ) - \lambda ( s ) }{t - s }
 +
} } \right | .
 +
$$
  
An important property is that the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020010.png" />-algebra associated with this topology coincides with the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020011.png" />-algebra.
+
The following distance, introduced by P. Billingsley [[#References|[a1]]], induces the Skorokhod topology and makes  $  D [ 0,1 ] $
 +
a complete separable [[Metric space|metric space]]:
  
The Skorokhod topology on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020012.png" /> of right-continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020013.png" /> having limits to the left can be defined by requiring the convergence in the Skorokhod metric on each compact interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020015.png" />.
+
$$
 +
d ( x,y ) = \inf  _ {\lambda \in \Lambda } \left \{ \left \| \lambda \right \| +  \sup  _ {t \in [ 0,1 ] } \left | {x ( t ) - x ( \lambda ( t ) ) } \right | \right \} .
 +
$$
  
Applying Prokhorov's theorem [[#References|[a3]]] to the complete separable metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020016.png" /> yields that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020018.png" />-valued random variables (cf. [[Random variable|Random variable]]) converges in distribution if and only if their finite-dimensional distributions converge and the laws of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020019.png" /> are tight (for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020020.png" /> there exists a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020023.png" />). Useful criteria for weak convergence can be deduced from this result and from the characterization of compact sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020024.png" /> (see [[#References|[a1]]]).
+
An important property is that the Borel  $  \sigma $-
 +
algebra associated with this topology coincides with the projection  $  \sigma $-
 +
algebra.
  
Complete separable distances on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020025.png" /> of functions with possible jumps on an arbitrary parameter set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020026.png" /> are introduced in [[#References|[a5]]], and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110200/s11020027.png" /> these distances have been applied to obtain criteria for the convergence in law of multi-parameter stochastic processes.
+
The Skorokhod topology on the space $  D [ 0, \infty ) $
 +
of right-continuous functions on  $  [ 0, \infty ) $
 +
having limits to the left can be defined by requiring the convergence in the Skorokhod metric on each compact interval  $  [ 0,T ] $,
 +
$  T > 0 $.
 +
 
 +
Applying Prokhorov's theorem [[#References|[a3]]] to the complete separable metric space  $  D [ 0, \infty ) $
 +
yields that a sequence  $  \{ X _ {n} \} $
 +
of  $  D [ 0, \infty ) $-
 +
valued random variables (cf. [[Random variable|Random variable]]) converges in distribution if and only if their finite-dimensional distributions converge and the laws of  $  \{ X _ {n} \} $
 +
are tight (for every  $  \epsilon > 0 $
 +
there exists a compact set  $  K \in D [ 0, \infty ) $
 +
such that  $  {\mathsf P} \{ X _ {n} \in K \} \geq  1 - \epsilon $
 +
for all  $  n $).
 +
Useful criteria for weak convergence can be deduced from this result and from the characterization of compact sets in  $  D [ 0, \infty ) $(
 +
see [[#References|[a1]]]).
 +
 
 +
Complete separable distances on the space  $  D ( T ) $
 +
of functions with possible jumps on an arbitrary parameter set $  T $
 +
are introduced in [[#References|[a5]]], and for $  T \subset  \mathbf R  ^ {k} $
 +
these distances have been applied to obtain criteria for the convergence in law of multi-parameter stochastic processes.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pollard,  "Convergence of stochastic processes" , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.V. Prokhorov,  "Convergence of random processes and limit theorems in probability theory"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 157–214</TD></TR><TR><TD valign="top">[a4]</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">[a5]</TD> <TD valign="top">  M.L. Straf,  "Weak convergence of stochastic processes with several parameters" , ''Proc. Sixth Berkeley Symp. Math. Stat. and Prob.''  (1972)  pp. 187–221</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Billingsley,  "Convergence of probability measures" , Wiley  (1968)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Pollard,  "Convergence of stochastic processes" , Springer  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y.V. Prokhorov,  "Convergence of random processes and limit theorems in probability theory"  ''Th. Probab. Appl.'' , '''1'''  (1956)  pp. 157–214</TD></TR><TR><TD valign="top">[a4]</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">[a5]</TD> <TD valign="top">  M.L. Straf,  "Weak convergence of stochastic processes with several parameters" , ''Proc. Sixth Berkeley Symp. Math. Stat. and Prob.''  (1972)  pp. 187–221</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


A topological structure (topology) on the space $ D [ 0,1 ] $ of right-continuous functions on $ [ 0,1 ] $ having limits to the left at each $ t \in ( 0,1 ] $, introduced by A.V. Skorokhod [a4] as an alternative to the topology of uniform convergence in order to study the convergence in distribution of stochastic processes with jumps.

Let $ \Lambda $ be the class of strictly increasing, continuous mappings of $ [ 0,1 ] $ onto itself. For $ \lambda \in \Lambda $ one defines

$$ \left \| \lambda \right \| = \sup _ {s \neq t } \left | { { \mathop{\rm log} } { \frac{\lambda ( t ) - \lambda ( s ) }{t - s } } } \right | . $$

The following distance, introduced by P. Billingsley [a1], induces the Skorokhod topology and makes $ D [ 0,1 ] $ a complete separable metric space:

$$ d ( x,y ) = \inf _ {\lambda \in \Lambda } \left \{ \left \| \lambda \right \| + \sup _ {t \in [ 0,1 ] } \left | {x ( t ) - x ( \lambda ( t ) ) } \right | \right \} . $$

An important property is that the Borel $ \sigma $- algebra associated with this topology coincides with the projection $ \sigma $- algebra.

The Skorokhod topology on the space $ D [ 0, \infty ) $ of right-continuous functions on $ [ 0, \infty ) $ having limits to the left can be defined by requiring the convergence in the Skorokhod metric on each compact interval $ [ 0,T ] $, $ T > 0 $.

Applying Prokhorov's theorem [a3] to the complete separable metric space $ D [ 0, \infty ) $ yields that a sequence $ \{ X _ {n} \} $ of $ D [ 0, \infty ) $- valued random variables (cf. Random variable) converges in distribution if and only if their finite-dimensional distributions converge and the laws of $ \{ X _ {n} \} $ are tight (for every $ \epsilon > 0 $ there exists a compact set $ K \in D [ 0, \infty ) $ such that $ {\mathsf P} \{ X _ {n} \in K \} \geq 1 - \epsilon $ for all $ n $). Useful criteria for weak convergence can be deduced from this result and from the characterization of compact sets in $ D [ 0, \infty ) $( see [a1]).

Complete separable distances on the space $ D ( T ) $ of functions with possible jumps on an arbitrary parameter set $ T $ are introduced in [a5], and for $ T \subset \mathbf R ^ {k} $ these distances have been applied to obtain criteria for the convergence in law of multi-parameter stochastic processes.

References

[a1] P. Billingsley, "Convergence of probability measures" , Wiley (1968)
[a2] D. Pollard, "Convergence of stochastic processes" , Springer (1984)
[a3] Y.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Th. Probab. Appl. , 1 (1956) pp. 157–214
[a4] A.V. Skorokhod, "Limit theorems for stochastic processes" Th. Probab. Appl. , 1 (1956) pp. 261–290
[a5] M.L. Straf, "Weak convergence of stochastic processes with several parameters" , Proc. Sixth Berkeley Symp. Math. Stat. and Prob. (1972) pp. 187–221
How to Cite This Entry:
Skorokhod topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skorokhod_topology&oldid=48733
This article was adapted from an original article by D. Nualart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article