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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a càdlàg (), 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 In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s (or a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
that admit (or even require) jumps, unlike
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
, 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 "", the left-right reversal of càdlàg, and càllàl for "" (continuous on one side, limit on the other side), for a function which at each point of the domain is either càdlàg or càglàd.


Definition

Let (M, d) be a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, and let E \subseteq \mathbb. A function f:E \to M is called a càdlàg function if, for every t \in E, * the left limit f(t-) := \lim_f(s) exists; and * the right limit f(t+) := \lim_f(s) exists and equals f(t). That is, f is right-continuous with left limits.


Examples

* All functions continuous on a subset of the real numbers are càdlàg functions on that subset. * As a consequence of their definition, all
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
s are càdlàg functions. For instance the cumulative at point r correspond to the probability of being lower or equal than r, namely \mathbb \leq r/math>. In other words, the semi-open interval of concern for a two-tailed distribution (-\infty, r] is right-closed. * The right derivative f^\prime_+ of any
convex function In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
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 \mathbb(E:M) (or simply \mathbb) and is called Skorokhod space after the Ukrainian mathematician Anatoliy Skorokhod. Skorokhod space can be assigned a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
that intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
only allows us to "wiggle space a bit"). For simplicity, take E = , T/math> and M = \mathbb^n — see Billingsley for a more general construction. We must first define an analogue of the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if :, f(x)-f(y), \leq\ ...
, \varpi'_ (\delta). For any F \subseteq E, set : w_ (F) := \sup_ , f(s) - f(t) , and, for \delta > 0, define the càdlàg modulus to be : \varpi'_ (\delta) := \inf_ \max_ w_ ( _, t_)), where the infimum runs over all partitions \Pi = \,\; k \in E, with \min_i(t_i - t_) > \delta. This definition makes sense for non-càdlàg f (just as the usual modulus of continuity makes sense for discontinuous functions). f is càdlàg if and only if \lim_ \varpi'_ (\delta) = 0. Now let \Lambda denote the set of all strictly increasing, continuous bijections from E to itself (these are "wiggles in time"). Let : \, f \, := \sup_ , f(t) , denote the uniform norm on functions on E. Define the Skorokhod metric \sigma on \mathbb by : \sigma (f, g) := \inf_ \max \, where I: E \to E is the identity function. In terms of the "wiggle" intuition, \, \lambda - I \, measures the size of the "wiggle in time", and \, f - g \circ \lambda \, measures the size of the "wiggle in space". The Skorokhod
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
is indeed a metric. The topology \Sigma generated by \sigma is called the Skorokhod topology on \mathbb. An equivalent metric, : d (f, g) := \inf_ (\, \lambda - I \, + \, f - g \circ \lambda \, ), was introduced independently and utilized in control theory for the analysis of switching systems.


Properties of Skorokhod space


Generalization of the uniform topology

The space C of continuous functions on E is a subspace of \mathbb. The Skorokhod topology relativized to C coincides with the uniform topology there.


Completeness

Although \mathbb is not a
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bo ...
with respect to the Skorokhod metric \sigma, there is a topologically equivalent metric \sigma_0 with respect to which \mathbb is complete.


Separability

With respect to either \sigma or \sigma_0, \mathbb is a
separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x_n )_^ of elements of the space such that every nonempty open subset of the space contains at least one elemen ...
. Thus, Skorokhod space is a
Polish space In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
.


Tightness in Skorokhod space

By an application of the
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded inte ...
, one can show that a sequence (\mu_n)_ of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s on Skorokhod space \mathbb is
tight Tight may refer to: Clothing * Skin-tight garment, a garment that is held to the skin by elastic tension * Tights, a type of leg coverings fabric extending from the waist to feet * Tightlacing, the practice of wearing a tightly-laced corset ...
if and only if both the following conditions are met: : \lim_ \limsup_ \mu_\big( \ \big) = 0, and : \lim_ \limsup_ \mu_\big( \ \big) = 0\text\varepsilon > 0.


Algebraic and topological structure

Under the Skorokhod topology and pointwise addition of functions, \mathbb is not a topological group, as can be seen by the following example: Let E=[0,2) be a half-open interval and take f_n = \chi_ \in \mathbb to be a sequence of characteristic functions. Despite the fact that f_n \rightarrow \chi_ in the Skorokhod topology, the sequence f_n - \chi_ does not converge to 0.


See also

*


References


Further reading

* * {{DEFAULTSORT:Cadlag Real analysis Stochastic processes