CÃ dlÃ g

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CÃ dlÃ g

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a càdlàg (French "continue à droite, limitée 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 number

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s (or a subset

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
of them) that is everywhere right-continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
and has left limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function near a particular independent variable....
s everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion

Brownian motion

Brownian motion is the seemingly random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a particle theory....
, which has continuous sample paths.

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In mathematics

Mathematics

Real number

Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
of them) that is everywhere right-continuous

Continuous function

Limit of a function

Brownian motion

Domain

Domain has several meanings:...
is known as Skorokhod space.

Definition

Let be a metric space

Metric space

In mathematics, a metric space is a Set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space....
, and let . A function is called a càdlàg function if, for every ,

the left limit exists; and
the right limit exists and equals .

That is, is right-continuous with left limits.

Examples

All continuous functions are càdlàg functions.
As a consequence of their definition, all cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
s are càdlàg functions.

Skorokhod space

The set of all càdlàg functions from to is often denoted by (or simply ) and is called Skorokhod space after the Ukrainian

Ukraine

Ukraine is a country in Eastern Europe. It is bordered by Russia to the east; Belarus to the north; Poland, Slovakia, and Hungary to the west; Romania and Moldova to the southwest; and the Black Sea and Sea of Azov to the south....
mathematician

Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
Anatoliy Skorokhod

Anatoliy Skorokhod

Anatoliy Volodymyrovych Skorokhod is a Ukraine mathematician, and an academician of the National Academy of Sciences of Ukraine since 1985.In 1956–1964 he worked at Kiev State University....
. Skorokhod space can be assigned a topology

Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
that, intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of uniform convergence

Uniform convergence

In the mathematics field of mathematical analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence of function converges uniformly to a limiting function f if the speed of convergence of fn to f does not depend on x....
only allows us to "wiggle space a bit"). For simplicity, take and — see Billingsley for a more general construction.

We must first define an analogue of the modulus of continuity

Modulus of continuity

In mathematics, the modulus of continuity is a precise way to measure the smoothness of a function. It is used as a delicate tool in mathematical analysis, to discuss highly non-smooth functions, which nonetheless enjoy some kind of smoothness....
, . For any , set

and, for , define the càdlàg modulus to be

where the infimum

Infimum

In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is less than or equal to all elements of the subset....
runs over all partitions , , with . This definition makes sense for non-càdlàg (just as the usual modulus of continuity makes sense for discontinuous functions) and it can be shown that is càdlàg if and only if

If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
as .

Now let denote the set of all strictly increasing, continuous bijection

Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
s from to itself (these are "wiggles in time"). Let

denote the uniform norm on functions on . Define the Skorokhod metric on by

,

where is the identity function. In terms of the "wiggle" intuition, measures the size of the "wiggle in time", and measures the size of the "wiggle in space".

It can be shown that the Skorokhod metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
is, indeed a metric. The topology generated by is called the Skorokhod topology on .

Properties of Skorokhod space

Generalization of the uniform topology

The space C of continuous functions on E is a subspace

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology ....
of D. The Skorokhod topology relativized to C coincides with the uniform topology there.

Completeness

It can be shown that, although D is not a complete space

Complete space

In mathematical analysis, a metric space M is said to be complete if every Cauchy sequence of points in M has a limit that is also in M or alternatively if every Cauchy sequence in M converges in M....
with respect to the Skorokhod metric σ, there is a topologically equivalent metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a Set . A set with a metric is called a metric space....
σ₀ with respect to which D is complete.

Separability

With respect to either σ or σ₀, D is a separable space

Separable space

In mathematics a topological space is called separable if it contains a countable set dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence....
. Thus, Skorokhod space is a Polish space

Polish space

In mathematics, a Polish space is a separable space complete space topological space; that is, a space homeomorphic to a Complete space metric space that has a countable Dense set subset....
.

Tightness in Skorokhod space

By an application of the Arzelà-Ascoli theorem

Arzelà-Ascoli theorem

In mathematics, the Arzel?Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether a set of continuous functions from a compact space metric space into a metric space is compact in the topological space of uniform convergence....
, one can show that a sequence of probability measures on Skorokhod space D is tight

Tightness of measures

In mathematics, tightness is a concept in measure theory, the intuitive idea being that a given collection of measures does not "escape to infinity."...
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.