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Càdlàg
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<h2 id="definition">Definition</h2> <p>Let ( M , d ) {\displaystyle (M,d)} be a <a href="/facts/Metric_space/gnBBRXtc">metric space</a>, and let E ⊆ R {\displaystyle E\subseteq \mathbb {R} } . A function f : E → M {\displaystyle f:E\to M} is called a càdlàg function if, for every t ∈ E {\displaystyle t\in E} , </p> <ul><li>the <a href="/facts/Left_limit/zLfr8gc5">left limit</a> f ( t − ) := lim s → t − f ( s ) {\displaystyle f(t-):=\lim _{s\to t^{-}}f(s)} exists; and</li> <li>the <a href="/facts/Right_limit/zLfr8gc5">right limit</a> f ( t + ) := lim s → t + f ( s ) {\displaystyle f(t+):=\lim _{s\to t^{+}}f(s)} exists and equals f ( t ) {\displaystyle f(t)} .</li></ul> <p>That is, f {\displaystyle f} is right-continuous with left limits. </p> <h2 id="examples">Examples</h2> <ul><li>All functions continuous on a subset of the real numbers are càdlàg functions on that subset.</li> <li>As a consequence of their definition, all <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution functions</a> are càdlàg functions. For instance the cumulative at point r {\displaystyle r} correspond to the probability of being lower or equal than r {\displaystyle r} , namely P [ X ≤ r ] {\displaystyle \mathbb {P} [X\leq r]} . In other words, the semi-open <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> of concern for a two-tailed distribution ( − ∞ , r ] {\displaystyle (-\infty ,r]} is right-closed.</li> <li>The right derivative f + ′ {\displaystyle f_{+}^{\prime }} of any <a href="/facts/Convex_function/IbzG5SLF">convex function</a> f {\displaystyle f} defined on an open interval, is an increasing cadlag function.</li></ul> <h2 id="skorokhod-space">Skorokhod space</h2> <p>The set of all càdlàg functions from E {\displaystyle E} to M {\displaystyle M} is often denoted by D ( E : M ) {\displaystyle \mathbb {D} (E:M)} (or simply D {\displaystyle \mathbb {D} } ) and is called Skorokhod space after the <a href="/facts/NASU_Institute_of_Mathematics/P2E7cx0l">Ukrainian mathematician</a> <a href="/facts/Anatoliy_Skorokhod/boIrt8wI">Anatoliy Skorokhod</a>. Skorokhod space can be assigned a <a href="/facts/Topology/2EqW47cU">topology</a> that intuitively allows us to "wiggle space and time a bit" (whereas the traditional topology of <a href="/facts/Uniform_convergence/ecz9eNWn">uniform convergence</a> only allows us to "wiggle space a bit").<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> For simplicity, take E = [ 0 , T ] {\displaystyle E=[0,T]} and M = R n {\displaystyle M=\mathbb {R} ^{n}} — see Billingsley<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> for a more general construction. </p><p>We must first define an analogue of the <a href="/facts/Modulus_of_continuity/31lThXhW">modulus of continuity</a>, ϖ f ′ ( δ ) {\displaystyle \varpi '_{f}(\delta )} . For any F ⊆ E {\displaystyle F\subseteq E} , set </p> w f ( F ) := sup s , t ∈ F | f ( s ) − f ( t ) | {\displaystyle w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|} <p>and, for δ > 0 {\displaystyle \delta >0} , define the càdlàg modulus to be </p> ϖ f ′ ( δ ) := inf Π max 1 ≤ i ≤ k w f ( [ t i − 1 , t i ) ) , {\displaystyle \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),} <p>where the <a href="/facts/Infimum/oXCKVopz">infimum</a> runs over all partitions Π = { 0 = t 0 < t 1 < ⋯ < t k = T } , k ∈ E {\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\},\;k\in E} , with min i ( t i − t i + 1 ) > δ {\displaystyle \min _{i}(t_{i}-t_{i+1})>\delta } . This definition makes sense for non-càdlàg f {\displaystyle f} (just as the usual modulus of continuity makes sense for discontinuous functions). f {\displaystyle f} is càdlàg <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> lim δ → 0 ϖ f ′ ( δ ) = 0 {\displaystyle \lim _{\delta \to 0}\varpi '_{f}(\delta )=0} . </p><p>Now let Λ {\displaystyle \Lambda } denote the set of all <a href="/facts/Strictly_increasing/MjQK0YL2">strictly increasing</a>, continuous <a href="/facts/Bijection/j4gTuTmW">bijections</a> from E {\displaystyle E} to itself (these are "wiggles in time"). Let </p> ‖ f ‖ := sup t ∈ E | f ( t ) | {\displaystyle \|f\|:=\sup _{t\in E}|f(t)|} <p>denote the uniform norm on functions on E {\displaystyle E} . Define the Skorokhod metric σ {\displaystyle \sigma } on D {\displaystyle \mathbb {D} } by </p> σ ( f , g ) := inf λ ∈ Λ max { ‖ λ − I ‖ , ‖ f − g ∘ λ ‖ } , {\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},} <p>where I : E → E {\displaystyle I:E\to E} is the identity function. In terms of the "wiggle" intuition, ‖ λ − I ‖ {\displaystyle \|\lambda -I\|} measures the size of the "wiggle in time", and ‖ f − g ∘ λ ‖ {\displaystyle \|f-g\circ \lambda \|} measures the size of the "wiggle in space". </p><p>The Skorokhod <a href="/facts/Metric_(mathematics)/RkUptSo8">metric</a> is indeed a metric. The topology Σ {\displaystyle \Sigma } generated by σ {\displaystyle \sigma } is called the Skorokhod topology on D {\displaystyle \mathbb {D} } . </p><p>An equivalent metric, </p> d ( f , g ) := inf λ ∈ Λ ( ‖ λ − I ‖ + ‖ f − g ∘ λ ‖ ) , {\displaystyle d(f,g):=\inf _{\lambda \in \Lambda }(\|\lambda -I\|+\|f-g\circ \lambda \|),} <p>was introduced independently and utilized in control theory for the analysis of switching systems.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="properties-of-skorokhod-space">Properties of Skorokhod space</h2> <h3>Generalization of the uniform topology</h3> <p>The space C {\displaystyle C} of continuous functions on E {\displaystyle E} is a <a href="/facts/Subspace_topology/NjU6z4Ca">subspace</a> of D {\displaystyle \mathbb {D} } . The Skorokhod topology relativized to C {\displaystyle C} coincides with the uniform topology there. </p> <h3>Completeness</h3> <p>Although D {\displaystyle \mathbb {D} } is not a <a href="/facts/Complete_space/lsckmU8G">complete space</a> with respect to the Skorokhod metric σ {\displaystyle \sigma } , there is a <a href="/facts/Metric_(mathematics)/RkUptSo8">topologically equivalent metric</a> σ 0 {\displaystyle \sigma _{0}} with respect to which D {\displaystyle \mathbb {D} } is complete.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h3>Separability</h3> <p>With respect to either σ {\displaystyle \sigma } or σ 0 {\displaystyle \sigma _{0}} , D {\displaystyle \mathbb {D} } is a <a href="/facts/Separable_space/2bmU73bk">separable space</a>. Thus, Skorokhod space is a <a href="/facts/Polish_space/CkUFcGap">Polish space</a>. </p> <h3>Tightness in Skorokhod space</h3> <p>By an application of the <a href="/facts/Arzel%25C3%25A0%25E2%2580%2593Ascoli_theorem/eqSFHXbo">Arzelà–Ascoli theorem</a>, one can show that a sequence ( μ n ) n = 1 , 2 , … {\displaystyle (\mu _{n})_{n=1,2,\dots }} of <a href="/facts/Probability_measure/8BjhgoPR">probability measures</a> on Skorokhod space D {\displaystyle \mathbb {D} } is <a href="/facts/Tightness_of_measures/K9dvSWDp">tight</a> if and only if both the following conditions are met: </p> lim a → ∞ lim sup n → ∞ μ n ( { f ∈ D | ‖ f ‖ ≥ a } ) = 0 , {\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\|f\|\geq a\}{\big )}=0,} <p>and </p> lim δ → 0 lim sup n → ∞ μ n ( { f ∈ D | ϖ f ′ ( δ ) ≥ ε } ) = 0 for all ε > 0. {\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}{\big (}\{f\in \mathbb {D} \;|\;\varpi '_{f}(\delta )\geq \varepsilon \}{\big )}=0{\text{ for all }}\varepsilon >0.} <h3>Algebraic and topological structure</h3> <p>Under the Skorokhod topology and pointwise addition of functions, D {\displaystyle \mathbb {D} } is not a topological group, as can be seen by the following example: </p><p>Let E = [ 0 , 2 ) {\displaystyle E=[0,2)} be a half-open interval and take f n = χ [ 1 − 1 / n , 2 ) ∈ D {\displaystyle f_{n}=\chi _{[1-1/n,2)}\in \mathbb {D} } to be a sequence of characteristic functions. Despite the fact that f n → χ [ 1 , 2 ) {\displaystyle f_{n}\rightarrow \chi _{[1,2)}} in the Skorokhod topology, the sequence f n − χ [ 1 , 2 ) {\displaystyle f_{n}-\chi _{[1,2)}} does not converge to 0. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Classical_Wiener_space/twJErlPb">Classical Wiener space</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Billingsley, Patrick (1995). <i>Probability and Measure</i>. New York, NY: John Wiley & Sons, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-00710-2.</li> <li>Billingsley, Patrick (1999). <a href="https://archive.org/details/convergenceofpro0000bill"><i>Convergence of Probability Measures</i></a>. New York, NY: John Wiley & Sons, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-19745-9.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Skorokhod space - Encyclopedia of Mathematics". <a href="https://encyclopediaofmath.org/wiki/Skorokhod_space" target="_blank">https://encyclopediaofmath.org/wiki/Skorokhod_space</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Billingsley, P. Convergence of Probability Measures. New York: Wiley. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Georgiou, T.T. and Smith, M.C. (2000). "Robustness of a relaxation oscillator". International Journal of Robust and Nonlinear Control. 10 (11–12): 1005–1024. doi:10.1002/1099-1239(200009/10)10:11/12<1005::AID-RNC536>3.0.CO;2-Q.{{cite journal}}: CS1 maint: multiple names: authors list (link) <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Billingsley, P. Convergence of Probability Measures. New York: Wiley. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>