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