View source for Skorokhod topology

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.

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

<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>

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]]:

<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>

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 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" />.

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]]]).

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.

====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>