# Skorokhod topology

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