World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0030862839
Reproduction Date:

Title: Càdlàg  
Author: World Heritage Encyclopedia
Language: English
Subject: Doob's martingale inequality, Path space, Wiener process, Local time (mathematics), Stochastic processes
Collection: Real Analysis, Stochastic Processes
Publisher: World Heritage Encyclopedia


In mathematics, a càdlàg (French "continue à droite, limite à gauche"), RCLL (“right continuous with left limits”), 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. Càdlàg 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 càdlàg functions on a given domain is known as Skorokhod space.

Two related terms are càglàd, standing for "continue à gauche, limite à droite", the left-right reversal of càdlàg, and càllàl 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 càdlàg or càglàd at each point of the domain.


  • Definition 1
  • Examples 2
  • Skorokhod space 3
  • Properties of Skorokhod space 4
    • Generalization of the uniform topology 4.1
    • Completeness 4.2
    • Separability 4.3
    • Tightness in Skorokhod space 4.4
    • Algebraic and topological structure 4.5
  • References 5


Cumulative distribution functions are examples of càdlàg functions.

Let (M, d) be a metric space, and let ER. A function ƒ: EM is called a càdlàg function if, for every tE,

  • the left limit ƒ(t−) := lims↑tƒ(s) exists; and
  • the right limit ƒ(t+) := lims↓tƒ(s) exists and equals ƒ(t).

That is, ƒ is right-continuous with left limits.


  • All continuous functions are càdlàg functions.
  • As a consequence of their definition, all cumulative distribution functions are càdlàg functions. For instance the cumulative at point r correspond to the probability of being lower or equal than r, namely \mathbb{P}[x\leq r]. In other words, the semi-open interval of concern for a two-tailed distribution (-\infty, r] is right-closed.
  • The right derivative f+' of any convex function f defined on an open interval, is an increasing cadlag function.

Skorokhod space

The set of all càdlàg 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 FE, set

w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |

and, for δ > 0, define the càdlàg modulus to be

\varpi'_{f} (\delta) := \inf_{\Pi} \max_{1 \leq i \leq k} w_{f} ([t_{i - 1}, t_{i})),

where the infimum runs over all partitions Π = {0 = t0 < t1 < … < tk = T}, kN, with mini (ti − ti−1) > δ. This definition makes sense for non-càdlàg ƒ (just as the usual modulus of continuity makes sense for discontinuous functions) and it can be shown that ƒ is càdlàg 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

\| f \| := \sup_{t \in E} | f(t) |

denote the uniform norm on functions on E. Define the Skorokhod metric σ on D by

\sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda - I \|, \| f - g \circ \lambda \| \},

where I: EE 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.


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.


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:

\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in D \;|\; \| f \| \geq a \} \big) = 0,


\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in D \;|\; \varpi'_{f} (\delta) \geq \varepsilon \} \big) = 0\text{ for all }\varepsilon > 0.

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 E=[0,2) be the unit interval and take f_n = \chi_{[1-1/n,2)} \in D to be a sequence of characteristic functions. Despite the fact that f_n \rightarrow \chi_{[1,2)} in the Skorokhod topology, the sequence f_n - \chi_{[1,2)} does not converge to 0.


  • Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc.  
  • Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.  

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.