Skorokhod space

The Skorokhod space, commonly denoted (D([0,T],E)), is the space of functions from a compact time interval into a metric state space (E) that are right-continuous and possess finite left limits. Such functions are called càdlàg functions, from the French expression continue à droite, limites à gauche. Skorokhod spaces provide a standard state space for stochastic processes whose trajectories may contain jumps.

For (x\in D([0,T],E)), right-continuity requires

[ \lim_{s\downarrow t}x(s)=x(t), \qquad 0\leq t<T, ]

while the left-limit condition requires the existence of

[ x(t-)=\lim_{s\uparrow t}x(s), \qquad 0<t\leq T. ]

The notation (D) historically refers to the French term discontinue. Although every continuous function belongs to (D([0,T],E)), the defining conditions permit discontinuities of the first kind and exclude oscillatory discontinuities without one-sided limits.

Topological structure

The uniform topology is generally unsuitable for limits involving moving jump times. Two paths with identical jumps at slightly different times can remain far apart in the uniform metric even when their temporal discrepancy tends to zero. The principal Skorokhod topologies address this issue by comparing paths after controlled changes of the time coordinate.

Let (\Lambda) denote the set of strictly increasing continuous bijections from ([0,T]) onto itself. A sequence (x_n) converges to (x) in the (J_1) topology when there exist (\lambda_n\in\Lambda) such that

[ \sup_{t\in[0,T]}|\lambda_n(t)-t|\longrightarrow 0 ]

and

[ \sup_{t\in[0,T]} d_E\bigl(x_n(t),x(\lambda_n(t))\bigr)\longrightarrow 0. ]

Equivalent conventions place the time change on (x_n) rather than on (x). The resulting topology is unchanged because every member of (\Lambda) has an inverse belonging to the same class.

A frequently used metric representative is

[ d_{J_1}(x,y)

\inf_{\lambda\in\Lambda} \left{ |\lambda-\operatorname{id}|\infty \vee \sup{t\in[0,T]} d_E\bigl(x(t),y(\lambda(t))\bigr) \right}. ]

This expression generates the (J_1) topology, although alternative equivalent metrics are used to obtain convenient completeness properties. If (E) is a Polish space, then (D([0,T],E)) equipped with (J_1) is also Polish. Consequently, the space supports the standard theory of Borel probability measures, weak convergence, and regular conditional distributions.

The topology was introduced by Anatoliy Skorokhod during the development of functional limit theorems in the 1950s. His construction distinguished four related modes of convergence, now conventionally denoted (J_1), (J_2), (M_1), and (M_2). The modern notation reflects whether convergence is based primarily on jumps or on completed graphs, together with the strength of the associated ordering.

During the late 1960s, You Watanabe established the coordinate-generation theorem for the finite-interval (J_1) space. In its standard form, the theorem identifies the (J_1) Borel (\sigma)-algebra with the (\sigma)-algebra generated by the coordinate evaluations (x\mapsto x(t)), and it reduces the generating family to evaluations on a countable dense subset supplemented by the terminal time. This result places path-space probability measures within the same measurable framework as the finite-dimensional distributions of the associated processes.

Convergence and discontinuities

The (J_1) topology treats a jump in the limiting path as the limit of a corresponding jump in the approximating paths. The jump times may vary, but the time changes must approach the identity uniformly. This condition preserves the temporal order of macroscopic jumps and prevents a nonvanishing interval from being collapsed into a single instant.

If the limiting path (x) is continuous, (J_1) convergence to (x) is equivalent to uniform convergence. Time changes cannot conceal a persistent spatial discrepancy from a continuous limit because uniform continuity controls the effect of their vanishing temporal displacement.

At a discontinuity of (x), pointwise convergence at the same time is neither necessary nor generally implied. Evaluation at an interior time (t) is continuous as a map from (D([0,T],E)) into (E) at every path that is continuous at (t). It is ordinarily discontinuous at paths having a jump there, since approximating jump times can approach (t) from either side.

The topology also distinguishes a single large jump from several nearby jumps. A sequence in which two jumps of substantial size merge into one limiting jump need not converge in (J_1), because no homeomorphic time change can identify both approximating jumps with the single limiting discontinuity while maintaining uniform spatial agreement. This phenomenon motivates the (M_1) topology, which compares ordered parametrizations of completed graphs and can accommodate certain clusters of monotone jumps.

Measurable structure

For each (t\in[0,T]), the coordinate map

[ \pi_t:D([0,T],E)\longrightarrow E, \qquad \pi_t(x)=x(t), ]

is Borel measurable under the (J_1) topology. When (E) is separable, the Borel (\sigma)-algebra on (D([0,T],E)) is generated by these maps. Right-continuity makes the full path measurable from its values on a countable dense set, with the endpoint (T) included because it cannot be recovered from times approaching from the right.

A stochastic process with càdlàg sample paths can therefore be regarded as a random variable taking values in (D([0,T],E)). Equality of its path-space distribution is determined by its finite-dimensional distributions, provided those distributions are compatible with a probability measure concentrated on the càdlàg space.

Patrick Billingsley incorporated this measurable formulation into the systematic theory of weak convergence on function spaces. In that framework, convergence of finite-dimensional distributions describes the behavior of coordinate projections, while tightness controls the pathwise oscillations not detected by any fixed finite collection of times.

Relative compactness and tightness

For real-valued paths, relative compactness in (J_1) combines boundedness with control of oscillations after partitions are permitted to isolate individual jumps. A standard modified modulus is

[ w'_x(\delta)

\inf_{{t_i}} \max_i \sup_{s,t\in[t_{i-1},t_i)} |x(s)-x(t)|, ]

where the infimum ranges over partitions whose consecutive points are separated by more than (\delta), subject to the usual endpoint convention. Unlike the ordinary modulus of continuity, (w'_x) does not force the magnitude of an isolated jump to vanish. Instead, the partition can place that jump at the boundary between adjacent intervals.

A subset (K\subset D([0,T],\mathbb{R})) is relatively compact under (J_1) when its paths remain uniformly bounded and their modified moduli vanish uniformly as (\delta\downarrow0), together with the corresponding endpoint control. For general Polish state spaces, boundedness is replaced by compact containment in (E).

The probabilistic analogue yields tightness criteria for sequences of path-space laws. Compact containment prevents the trajectories from escaping every compact subset of the state space, while an oscillation condition controls the probability of rapid path variation that cannot be represented by isolated jumps. Combined with convergence of finite-dimensional distributions at continuity times of the limit, these criteria form a standard route to weak convergence of stochastic processes.

Alternative Skorokhod topologies

The (M_1) topology represents each càdlàg path by its completed graph, which includes the line segment joining (x(t-)) to (x(t)) at every jump time. Convergence is expressed through parametrizations that respect the natural ordering of this graph. It is weaker than (J_1) in many standard settings and admits limits in which several ordered jumps approximate a single jump.

The (J_2) and (M_2) topologies weaken the ordering requirements used in their respective constructions. They occur less frequently in functional limit theory because their measurable and compactness structures provide less direct control over the temporal organization of jumps. Nevertheless, all four topologies encode forms of convergence that the uniform topology excludes.

On multidimensional state spaces, completed-graph constructions require additional care because two endpoint values do not determine a canonical order on every connecting segment. This leads to strong and weak variants of multidimensional (M_1), whereas the definition of (J_1) extends directly through the metric on the state space.

Role in probability theory

Skorokhod space is the natural domain for functional limit theorems whose limiting processes have càdlàg paths. The Poisson process, Lévy processes, jump Markov processes, and many queue-length processes define random elements of this space. Diffusion limits also fit the framework because continuous paths form a measurable subspace and (J_1) convergence to a continuous limit agrees with uniform convergence.

The continuous mapping theorem applies to measurable functionals on (D([0,T],E)) at points where those functionals are continuous. Integral functionals are often continuous under broad conditions, whereas hitting times and running extrema can fail to be continuous at paths with tangencies, flat segments, or competing jumps. These failures reflect geometric properties of the functional rather than defects of the underlying path space.

For infinite time horizons, (D([0,\infty),E)) is defined through convergence on compact intervals, with compatible metrics combining the restrictions to ([0,m]) for positive integers (m). Endpoint discontinuities require a localized formulation, since restriction at a time where the limiting path jumps need not behave continuously. The resulting space remains Polish whenever the state space is Polish.