Cà dlà g
In mathematics, a cà dlà g (French "continue à droite, limite à gauche"), RCLL (“right continuous with left limitsâ€), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Cà dlà g functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of cà dlà g functions on a given domain is known as Skorokhod space.
Two related terms are cà glà d, standing for "continue à gauche, limite à droite", the left-right reversal of cà dlà g, and cà llà l for "continue à l'un, limite à l’autre" (continuous on one side, limit on the other side), for a function which is interchangeably either cà dlà g or cà glà d at each point of the domain. A more mellifluous English term is ricowil or, more whimsically, ricowilli, both terms standing for "right continuous with left limits".
Definition

Let (M, d) be a metric space, and let E ⊆ R. A function ƒ: E → M is called a cà dlà g function if, for every t ∈ E,
- the left limit ƒ(t−) := lims↑t ƒ(s) exists; and
- the right limit ƒ(t+) := lims↓t ƒ(s) exists and equals ƒ(t).
That is, Æ’ is right-continuous with left limits.
Examples
- All continuous functions are cà dlà g functions.
- As a consequence of their definition, all cumulative distribution functions are cà dlà g functions. For instance the cumulative at point
correspond to the probability of being lower or equal than
, namely
. In other words, the semi-open interval of concern for a two-tailed distribution
is right-closed.
- The right derivative
of any convex function f defined on an open interval, is an increasing cadlag function.
Skorokhod space
The set of all cà dlà g functions from E to M is often denoted by D(E; M) (or simply D) and is called Skorokhod space after the Soviet mathematician Anatoliy Skorokhod. Skorokhod space can be assigned a topology that, intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of uniform convergence only allows us to "wiggle space a bit"). For simplicity, take E = [0, T] and M = Rn — see Billingsley for a more general construction.
We must first define an analogue of the modulus of continuity, ϖ′ƒ(δ). For any F ⊆ E, set
and, for δ > 0, define the cà dlà g modulus to be
where the infimum runs over all partitions Π= {0 = t0 < t1 < … < tk = T}, k ∈ N, with mini (ti − ti−1) > δ. This definition makes sense for non-cà dlà g ƒ (just as the usual modulus of continuity makes sense for discontinuous functions) and it can be shown that ƒ is cà dlà g if and only if ϖ′ƒ(δ) → 0 as δ → 0.
Now let Λ denote the set of all strictly increasing, continuous bijections from E to itself (these are "wiggles in time"). Let
denote the uniform norm on functions on E. Define the Skorokhod metric σ on D by
where I: E → E is the identity function. In terms of the "wiggle" intuition, ||λ − I|| measures the size of the "wiggle in time", and ||ƒ − g○λ|| measures the size of the "wiggle in space".
It can be shown that the Skorokhod metric is indeed a metric. The topology Σ generated by σ is called the Skorokhod topology on D.
Properties of Skorokhod space
Generalization of the uniform topology
The space C of continuous functions on E is a subspace of D. The Skorokhod topology relativized to C coincides with the uniform topology there.
Completeness
It can be shown (Convergence of probability measures - Billingsley 1999) that, although D is not a complete space with respect to the Skorokhod metric σ, there is a topologically equivalent metric σ0 with respect to which D is complete.
Separability
With respect to either σ or σ0, D is a separable space. Thus, Skorokhod space is a Polish space.
Tightness in Skorokhod space
By an application of the Arzelà –Ascoli theorem, one can show that a sequence (μn)n=1,2,… of probability measures on Skorokhod space D is tight if and only if both the following conditions are met:
and
Algebraic and topological structure
Under the Skorokhod topology and pointwise addition of functions, D is not a topological group, as can be seen by the following example:
Let be the unit interval and take
to be a sequence of characteristic functions.
Despite the fact that
in the Skorokhod topology, the sequence
does not converge to 0.
References
- Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
- Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.