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2014-07-17T22:10:24Z

Task 2: Fix CS1 deprecated coauthor parameter errors

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en>Nisse3333: /* Basic step */ changed a subscript after second equation

2014-02-10T22:07:56Z

Basic step: changed a subscript after second equation

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2012-08-31T01:02:08Z

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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 number]]s (or a [[subset]] of them) that is everywhere [[right-continuous]] and has left [[Limit of a function|limit]]s 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 of a function|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==
[[Image:Discrete probability distribution illustration.png|right|thumb|[[Cumulative distribution functions]] are examples of càdlàg functions.]]

Let {{nowrap|(''M'', ''d'')}} be a [[metric space]], and let {{nowrap|''E'' ⊆ '''R'''}}. A function {{nowrap|''ƒ'': ''E'' → ''M''}} is called a '''càdlàg function''' if, for every {{nowrap|''t'' ∈ ''E''}},
* the [[left limit]] {{nowrap|''ƒ''(''t−'') :{{=}} lim''s↑t'' ''ƒ''(''s'')}} exists; and
* the [[right limit]] {{nowrap|''ƒ''(''t+'') :{{=}} lim''s↓t'' ''ƒ''(''s'')}} exists and equals ''ƒ''(''t'').
That is, ''ƒ'' is right-continuous with left limits.

==Examples==

* All continuous functions are càdlàg functions.
* As a consequence of their definition, all [[cumulative distribution function]]s are càdlàg functions.
* 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 {{nowrap|''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 {{nowrap|''E'' {{=}} [0, ''T'']}} and {{nowrap|''M'' {{=}} '''R'''''n''}} — see Billingsley for a more general construction.

We must first define an analogue of the [[modulus of continuity]], {{nowrap|''ϖ′ƒ''(''δ'')}}. For any {{nowrap|''F'' ⊆ ''E''}}, set
: <math>
w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |
</math>
and, for {{nowrap|''δ'' > 0}}, define the '''càdlàg modulus''' to be
: <math>
\varpi'_{f} (\delta) := \inf_{\Pi} \max_{1 \leq i \leq k} w_{f} ([t_{i - 1}, t_{i})),
</math>
where the [[infimum]] runs over all partitions {{nowrap|Π {{=}} {0 {{=}} ''t''0 < ''t''1 < … < ''tk {{=}} T''}}}, {{nowrap|''k'' ∈ '''N'''}}, with {{nowrap|min''i'' (''ti − t''''i''−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]] {{nowrap|''ϖ′ƒ''(''δ'') → 0}} as {{nowrap|''δ'' → 0}}.

Now let Λ denote the set of all [[strictly increasing]], continuous [[bijection]]s from ''E'' to itself (these are "wiggles in time"). Let
: <math>
\| f \| := \sup_{t \in E} | f(t) |
</math>
denote the uniform norm on functions on ''E''. Define the '''Skorokhod metric''' ''σ'' on ''D'' by
: <math>
\sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda - I \|, \| f - g \circ \lambda \| \},
</math>
where {{nowrap|''I'': ''E'' → ''E''}} is the identity function. In terms of the "wiggle" intuition, {{nowrap|{{!!}}''λ − I''{{!!}}}} measures the size of the "wiggle in time", and {{nowrap|{{!!}}''ƒ − g○λ''{{!!}}}} measures the size of the "wiggle in space".

It can be shown that the Skorokhod [[Metric (mathematics)|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 topology|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 [[Metric (mathematics)#Equivalence of metrics|topologically equivalent metric]] ''σ''0 with respect to which ''D'' is complete.

===Separability===

With respect to either ''σ'' or ''σ''0, ''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 measure]]s on Skorokhod space ''D'' is [[tightness of measures|tight]] if and only if both the following conditions are met:
: <math>
\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in D \;|\; \| f \| \geq a \} \big) = 0,
</math>
and
: <math>
\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.
</math>

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

==References==

* {{cite book | author=Billingsley, Patrick | title=Probability and Measure | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1995 | isbn=0-471-00710-2}}
* {{cite book | author=Billingsley, Patrick | title=Convergence of Probability Measures | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1999 | isbn=0-471-19745-9}}
[[Category:Real analysis]]
[[Category:Stochastic processes]]

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