Skorokhod space

Let $\mathcal{D} = \mathcal{D} [ 0,1 ]$ be the space of real-valued functions $x$ on $[ 0,1 ]$ that are right-continuous and have left-hand limits, i.e.

\begin{equation*} x ( t + ) = \operatorname { lim } _ { s \downarrow t } x ( s ) \ \text{exists}, \end{equation*}

\begin{equation*} x ( t + ) = x ( t ) \text { for all } \ 0 \leq t < 1 ,\, x ( t - ) = \operatorname { lim } _ { s \uparrow t } x ( s ) \text { exists for all } 0 < t \leq 1. \end{equation*}

(In probabilistic literature, such a function is also said to be a cadlag function, "cadlag" being an acronym for the French "continu à droite, limites à gauche" .) Introducing a norm on $\mathcal{D}$ by setting $\| x \| = \operatorname { sup } _ { 0 \leq t \leq 1} | x ( t ) |$, then $\mathcal{D}$ becomes a Banach space, but it is easy to see that it is non-separable (cf. also Separable space). This non-separability causes well-known problems of measurability in the theory of weak convergence of measures on the space. To overcome this inconvenience, A.V. Skorokhod introduced a metric (and topology) under which the space $\mathcal{D}$ becomes a separable metric space. Although the original metric introduced by Skorokhod has a drawback in the sense that the metric space obtained is not complete, it turned out (see [a6]) that it is possible to construct an equivalent metric (i.e., giving the same topology) under which the space $\mathcal{D}$ becomes a separable and complete metric space (sometimes, for such a metric space the term Polish space is used). This metric is defined as follows.

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

\begin{equation*} \| \lambda \| = \operatorname { sup } _ { 0 \leq s < t \leq 1 } | \operatorname { log } \{ ( t - s ) ^ { - 1 } ( \lambda ( t ) - \lambda ( s ) ) \} |. \end{equation*}

Then for $x , y \in \mathcal{D}$ one defines

\begin{equation*} d ( x , y ) = \operatorname { inf } _ { \lambda \in \Lambda } \operatorname { max } \left\{ \| \lambda \| , \operatorname { sup } _ { 0 \leq t \leq 1 } | x ( t ) - y ( \lambda ( t ) ) | \right\}. \end{equation*}

The topology generated by this metric is called the Skorokhod topology and the complete separable metric space $\mathcal{D}$ is called the Skorokhod space (cf. also Skorokhod topology). This space is very important in the theory of random processes (cf. also Stochastic process). The general theory of weak convergence of probability measures on metric spaces and, in particular, on the space $\mathcal{D}$ is well developed. This theory was started in the fundamental papers [a4], [a6], [a10], [a11]. A well-known reference on these topics is [a1] (see also the updated second edition [a2]).

Generalizations.

Several generalizations of the Skorokhod space are worth mentioning. Instead of real-valued functions on $[ 0,1 ]$ it is possible to consider functions defined on $[ 0 , \infty )$ and taking values in a metric space $E$. The space of cadlag functions obtained in this way is denoted by ${\cal{D}} _ { E } [ 0 , \infty )$ and if $E$ is a Polish space, then ${\cal{D}} _ { E } [ 0 , \infty )$, with the appropriate topology, is also a Polish space, see [a7] and [a9], where these spaces are treated systematically.

Another generalization is obtained when the one-dimensional parameter $t$ (often regarded as "time" ) is replaced by multi-variate variable $t = ( t _ { 1 } , \dots , t _ { k } )$. Let $[ 0,1 ] ^ { k }$ denote unit cube in $\mathbf{R} ^ { k }$. It is possible to introduce (see [a8], [a12]) the space $\mathcal{D} [ 0,1 ] ^ { k }$ of cadlag functions on $[ 0,1 ] ^ { k }$ and a Skorokhod topology on it. In [a3] one can find references and recent results on weak convergence of probability measures and, in particular, on the central limit theorem in this space.

In his fundamental paper [a11], Skorokhod introduced four topologies $M _ { 1 }$, $M _ { 2 }$, $J _ { 1 }$, $J_2$; the topology $J _ { 1 }$ became the most famous one and now bears his name (cf. also Skorokhod topology). At the end of the 1980s it was found that in certain problems the other topologies introduced by Skorokhod in the space of cadlag functions can be useful (see, for example, [a5], [a13], [a14]).

References

[a1]	P. Billingsley, "Convergence of probability measures" , Wiley (1968)
[a2]	P. Billingsley, "Convergence of probability measures" , Wiley (1999) (Edition: Second)
[a3]	M. Bloznelis, V. Paulauskas, "On the central limit theorem for multiparameter stochastic processes" J. Hoffmann-Jorgensen (ed.) M. Marcus AND J. Kuelbs (ed.) , Probability in Banach spaces 9 (Proc. Conf.) , Birkhäuser (1994) pp. 155–172
[a4]	N.N. Chentsov, "Weak convergence of stochastic processes whose trajectories have no discontinuities of the second order" Th. Probab. Appl. , 1–3 (1956) pp. 140–143
[a5]	A. Jakubowski, "A non-Skorohod topology on the Skorohod space" Electron. J. Probab. , 2 : 4 (1997) pp. 1–21
[a6]	A.N. Kolmogorov, "On Skorohod convergence" Th. Probab. Appl. , 1–3 (1956) pp. 213–222
[a7]	S.N. Ethier, T.G. Kurtz, "Markov processes: Characterization and convergence" , Wiley (1986)
[a8]	G. Neuhaus, "On weak convergence of stochastic processes with multidimensional time parameter" Ann. Math. Stat. , 42 (1971) pp. 1285–1295
[a9]	D. Pollard, "Convergence of stochastic processes" , Springer (1984)
[a10]	Yu.V. Prokhorov, "Convergence of random processes and limit theorems in probability theory" Th. Probab. Appl. , 1–3 (1956) pp. 157–214
[a11]	A.V. Skorokhod, "Limit theorems for stochastic processes" Th. Probab. Appl. , 1–3 (1956) pp. 261–290
[a12]	M.L. Straf, "Weak convergence of stochastic processes with several parameters" Proc. Sixth Berkeley Symp. Math. Statist. Probab. , 2 (1972) pp. 187–221
[a13]	F. Avram, M. Taqqu, "Probability bounds for $M$-Skorokhod oscillations" Stochastic Processes Appl. , 33 (1989) pp. 63–72
[a14]	F. Avram, M. Taqqu, "Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction" Ann. Probab. , 20 (1992) pp. 483–503