https://en.formulasearchengine.com/api.php?action=feedcontributions&user=78.13.117.226&feedformat=atom formulasearchengine - User contributions [en] 2021-04-17T19:33:58Z User contributions MediaWiki 1.37.0-alpha https://en.formulasearchengine.com/index.php?title=C%C3%A0dl%C3%A0g&diff=268015 Càdlàg 2012-07-29T02:12:14Z <p>78.13.117.226: </p> <hr /> <div>In [[mathematics]], a '''càdlàg''' (French &quot;continue à droite, limite à gauche&quot;), '''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'''.<br /> <br /> Two related terms are '''càglàd''', standing for &quot;continue à gauche, limite à droite&quot;, the left-right reversal of càdlàg, and '''càllàl''' for&quot;continue à l'un, limite à l’autre&quot; (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.<br /> <br /> ==Definition==<br /> [[Image:Discrete probability distribution illustration.png|right|thumb|[[Cumulative distribution functions]] are examples of càdlàg functions.]]<br /> <br /> Let {{nowrap|(''M'', ''d'')}} be a [[metric space]], and let {{nowrap|''E'' ⊆ '''R'''}}. A function {{nowrap|''ƒ'':&amp;thinsp;''E'' → ''M''}} is called a '''càdlàg function''' if, for every {{nowrap|''t'' ∈ ''E''}},<br /> * the [[left limit]] {{nowrap|''ƒ''(''t−'') :{{=}} lim&lt;sub&gt;''s↑t''&lt;/sub&gt;&amp;thinsp;''ƒ''(''s'')}} exists; and<br /> * the [[right limit]] {{nowrap|''ƒ''(''t+'') :{{=}} lim&lt;sub&gt;''s↓t''&lt;/sub&gt;&amp;thinsp;''ƒ''(''s'')}} exists and equals ''ƒ''(''t'').<br /> That is, ''ƒ'' is right-continuous with left limits.<br /> <br /> ==Examples==<br /> <br /> * All continuous functions are càdlàg functions.<br /> * As a consequence of their definition, all [[cumulative distribution function]]s are càdlàg functions.<br /> * The right derivative ''f&lt;sub&gt;+&lt;/sub&gt;''' of any [[convex function]] '' f'' defined on an open interval, is an increasing cadlag function.<br /> <br /> ==Skorokhod space==<br /> <br /> 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 [[Ukraine|Ukrainian]] [[mathematician]] [[Anatoliy Skorokhod]]. Skorokhod space can be assigned a [[topology]] that, intuitively allows us to&quot;wiggle space and time a bit&quot; (whereas the traditional topology of [[uniform convergence]] only allows us to &quot;wiggle space a bit&quot;). For simplicity, take {{nowrap|''E'' {{=}} [0, ''T'']}} and {{nowrap|''M'' {{=}} '''R'''&lt;sup&gt;''n''&lt;/sup&gt;}} — see Billingsley for a more general construction.<br /> <br /> We must first define an analogue of the [[modulus of continuity]], {{nowrap|''ϖ′&lt;sub&gt;ƒ&lt;/sub&gt;''(''δ'')}}. For any {{nowrap|''F'' ⊆ ''E''}}, set<br /> : &lt;math&gt;<br /> w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |<br /> &lt;/math&gt;<br /> and, for {{nowrap|''δ'' &gt; 0}}, define the '''càdlàg modulus''' to be<br /> : &lt;math&gt;<br /> \varpi'_{f} (\delta) := \inf_{\Pi} \max_{1 \leq i \leq k} w_{f} ([t_{i - 1}, t_{i})),<br /> &lt;/math&gt;<br /> where the [[infimum]] runs over all partitions {{nowrap|Π {{=}} {0 {{=}} ''t''&lt;sub&gt;0&lt;/sub&gt; &lt; ''t''&lt;sub&gt;1&lt;/sub&gt; &lt; … &lt; ''t&lt;sub&gt;k&lt;/sub&gt; {{=}} T''}}}, {{nowrap|''k'' ∈ '''N'''}}, with {{nowrap|min&lt;sub&gt;''i''&lt;/sub&gt;&amp;thinsp;(''t&lt;sub&gt;i&lt;/sub&gt; − t''&lt;sub&gt;''i''−1&lt;/sub&gt;) &gt; ''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′&lt;sub&gt;ƒ&lt;/sub&gt;''(''δ'') → 0}} as {{nowrap|''δ'' → 0}}.<br /> <br /> Now let Λ denote the set of all [[strictly increasing]], continuous [[bijection]]s from ''E'' to itself (these are &quot;wiggles in time&quot;). Let<br /> : &lt;math&gt;<br /> \| f \| := \sup_{t \in E} | f(t) |<br /> &lt;/math&gt;<br /> denote the uniform norm on functions on ''E''. Define the '''Skorokhod metric''' ''σ'' on ''D'' by<br /> : &lt;math&gt;<br /> \sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda - I \|, \| f - g \circ \lambda \| \},<br /> &lt;/math&gt;<br /> where {{nowrap|''I'':&amp;thinsp;''E'' → ''E''}} is the identity function. In terms of the&quot;wiggle&quot; intuition, {{nowrap|{{!!}}''λ − I''{{!!}}}} measures the size of the&quot;wiggle in time&quot;, and {{nowrap|{{!!}}''ƒ − g○λ''{{!!}}}} measures the size of the&quot;wiggle in space&quot;.<br /> <br /> 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''.<br /> <br /> ==Properties of Skorokhod space==<br /> <br /> ===Generalization of the uniform topology===<br /> <br /> 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.<br /> <br /> ===Completeness===<br /> <br /> 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''&lt;sub&gt;0&lt;/sub&gt; with respect to which ''D'' is complete.<br /> <br /> ===Separability===<br /> <br /> With respect to either ''σ'' or ''σ''&lt;sub&gt;0&lt;/sub&gt;, ''D'' is a [[separable space]]. Thus, Skorokhod space is a [[Polish space]].<br /> <br /> ===Tightness in Skorokhod space===<br /> By an application of the [[Arzelà–Ascoli theorem]], one can show that a sequence (''μ&lt;sub&gt;n&lt;/sub&gt;'')&lt;sub&gt;''n''=1,2,…&lt;/sub&gt; of [[probability measure]]s on Skorokhod space ''D'' is [[tightness of measures|tight]] if and only if both the following conditions are met:<br /> : &lt;math&gt;<br /> \lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n}\big( \{ f \in D \;|\; \| f \| \geq a \} \big) = 0,<br /> &lt;/math&gt;<br /> and<br /> : &lt;math&gt;<br /> \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 &gt; 0.<br /> &lt;/math&gt;<br /> <br /> ===Algebraic and topological structure===<br /> Under the Skorokhod topology and pointwise addition of functions, ''D'' is not a topological group, as can be seen by the following example:<br /> <br /> Let &lt;math&gt; E=[0,2) &lt;/math&gt; be the unit interval and take &lt;math&gt;f_n = \chi_{[1/n,2)} \in D&lt;/math&gt; to be a sequence of characteristic functions.<br /> Despite the fact that &lt;math&gt; f_n \rightarrow \chi_{[1,2)} &lt;/math&gt; in the Skorokhod topology, the sequence &lt;math&gt; f_n - \chi_{[1,2)} &lt;/math&gt; does not converge to 0.<br /> <br /> ==References==<br /> <br /> * {{cite book | author=Billingsley, Patrick | title=Probability and Measure | publisher=John Wiley &amp; Sons, Inc. | location=New York, NY | year=1995 | isbn=0-471-00710-2}}<br /> * {{cite book | author=Billingsley, Patrick | title=Convergence of Probability Measures | publisher=John Wiley &amp; Sons, Inc. | location=New York, NY | year=1999 | isbn=0-471-19745-9}}<br /> [[Category:Real analysis]]<br /> [[Category:Stochastic processes]]<br /> <br /> {{DEFAULTSORT:Cadlag}}<br /> <br /> [[de:Càdlàg]]<br /> [[fr:Càdlàg]]<br /> [[it:Funzione càdlàg]]<br /> [[hu:Càdlàg]]<br /> [[uk:Неперервна справа функція з лівосторонніми границями]]<br /> [[vi:Càdlàg]]<br /> [[zh:右连左极函数]]</div> 78.13.117.226