右连左极函数

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发表于 2016-10-8 05:15:31 1.6万 4 | 显示全部楼层 |倒序浏览

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在数学中，右连左极函数（càdlàg，RCLL）是指定义在实数集或其子集上的处处右连续且有左极限的函数。这类函数在研究有跳跃甚至是需要跳跃的随机过程时很重要，这类随机过程不像布朗运动具有连续的样本轨道。给定定义域上的右连左极函数的集合称为斯科罗霍德空间（Skorokhod space）。
[micxp_threadbk]
[micxp_title]
定义
例子
斯科罗霍德空间
斯科罗霍德空间的性质
完备性
分离性
斯科罗霍德空间中的胎紧性
代数结构与拓扑结构
参考文献
[/micxp_title]
[#]

<img alt="" src="data/attachment/baike/375483/1476172446.png" width="220" height="248" class="thumbimage" />

累积分布函数是右连左极函数的一个例子。

令<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo stretchy="false">(</mo>
 <mi>M</mi>
 <mo>,</mo>
 <mi>d</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle (M,d)}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78e6f2ddf5baee227ee2a9f164726ba0c23c263" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.554ex; height:2.843ex;" alt="(M,d)" />为度量空间，并令<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>E</mi>
 <mo>⊆</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi mathvariant="double-struck">R</mi>
 </mrow>
 </mstyle>
</mrow>
{\displaystyle E\subseteq \mathbb {R} }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651e39fa02a0a98bc7f719fb35a883abe09bd8c5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.584ex; height:2.509ex;" alt="E \subseteq \mathbb{R}" />。函数<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 <mo>:</mo>
 <mi>E</mi>
 <mo stretchy="false">→</mo>
 <mi>M</mi>
 </mstyle>
</mrow>
{\displaystyle f:E\to M}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256ed3511e25ca61629b3ffa71176666dde9f6b8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:11.1ex; height:2.509ex;" alt="f : E \to M" />称为右连左极函数。若对于每一<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>t</mi>
 <mo>∈</mo>
 <mi>E</mi>
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</mrow>
{\displaystyle t\in E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e440b636de9fe0ade6ae59c02c2de5b3130d8cf8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.487ex; height:2.176ex;" alt="t \in E" />，都有

<ul>
<li>左极限<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>t</mi>
 <mo>−</mo>
 <mo stretchy="false">)</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">lim</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>s</mi>
 <mo stretchy="false">↑</mo>
 <mi>t</mi>
 </mrow>
 </munder>
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>s</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle f(t-):=\lim _{s\uparrow t}f(s)}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c764ac3690bde44ebe1e24e1595b34b66066a75c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:17.422ex; height:4.343ex;" alt="f(t-) := \lim_{s \uparrow t} f(s)" />存在；且</li>
<li>右极限<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>t</mi>
 <mo>+</mo>
 <mo stretchy="false">)</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">lim</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>s</mi>
 <mo stretchy="false">↓</mo>
 <mi>t</mi>
 </mrow>
 </munder>
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>s</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle f(t+):=\lim _{s\downarrow t}f(s)}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bed0c185fe8ea7dfa09617eb92a866b0454a4ee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:17.422ex; height:4.343ex;" alt="f(t+) := \lim_{s \downarrow t} f(s)" />存在并等于<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>t</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle f(t)}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:3.969ex; height:2.843ex;" alt="f(t)" />，</li>
</ul>

即<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 </mstyle>
</mrow>
{\displaystyle f}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.289ex; height:2.509ex;" alt="f" /> 是右连续的且有左极限。

[##]

<ul>
<li>全部连续函数都是右连左极函数。</li>
<li>由累积分布函数的定义知所有的累积分布函数都是右连左极函数。</li>
</ul>
[###]

从<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>E</mi>
 </mstyle>
</mrow>
{\displaystyle E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.786ex; height:2.176ex;" alt="E" />到<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>M</mi>
 </mstyle>
</mrow>
{\displaystyle M}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="M" />的所有右连左极函数的集合常记为<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>D</mi>
 <mo stretchy="false">(</mo>
 <mi>E</mi>
 <mo>;</mo>
 <mi>M</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle D(E;M)}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c608628c591703afa3fc202ab0f8d8f3452dfe" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="D(E; M)" />或简记为<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>D</mi>
 </mstyle>
</mrow>
{\displaystyle D}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.935ex; height:2.176ex;" alt="D" />，称为斯科罗霍德空间，是以乌克兰数学家阿纳托利·斯科罗霍德（Anatoliy Skorokhod）的名字命名。斯科罗霍德空间可以被指派一个拓扑，这一拓扑直觉上能使我们“稍微蠕动空间和时间”（而传统的一致收敛拓扑仅允许我们“稍微蠕动空间”）。为了简化说明，取<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>E</mi>
 <mo>=</mo>
 <mo stretchy="false">[</mo>
 <mn>0</mn>
 <mo>,</mo>
 <mi>T</mi>
 <mo stretchy="false">]</mo>
 </mstyle>
</mrow>
{\displaystyle E=[0,T]}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/854c8a073e60092dd8089b8b3e11d4272ba64d2d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:10.074ex; height:2.843ex;" alt="E = [0, T]" />，<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>M</mi>
 <mo>=</mo>
 <msup>
 <mrow class="MJX-TeXAtom-ORD">
 <mi mathvariant="double-struck">R</mi>
 </mrow>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 </mrow>
 </msup>
 </mstyle>
</mrow>
{\displaystyle M=\mathbb {R} ^{n}}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98351dc4e47f515338238b4eed6be6d46357bb5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.476ex; height:2.343ex;" alt="M = \mathbb{R}^{n}" />（Billingsley的书中描述了更一般的拓扑）

首先我们必须定义连续性模的一个模拟<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <msubsup>
 <mi>ϖ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 <mo>′</mo>
 </msubsup>
 <mo stretchy="false">(</mo>
 <mi>δ</mi>
 <mo stretchy="false">)</mo>
 </mstyle>
</mrow>
{\displaystyle \varpi '_{f}(\delta )}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dd6cc11360b1ebb555827b44fef0b420ba3387" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:5.968ex; height:3.343ex;" alt="\varpi'_{f} (\delta)" />。对于任意<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>F</mi>
 <mo>⊆</mo>
 <mi>E</mi>
 </mstyle>
</mrow>
{\displaystyle F\subseteq E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3400f80a7838cc279728b12a6e859ca65e353c6f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.646ex; height:2.509ex;" alt="F \subseteq E" />，使

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <msub>
 <mi>w</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 </msub>
 <mo stretchy="false">(</mo>
 <mi>F</mi>
 <mo stretchy="false">)</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">sup</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>s</mi>
 <mo>,</mo>
 <mi>t</mi>
 <mo>∈</mo>
 <mi>F</mi>
 </mrow>
 </munder>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>s</mi>
 <mo stretchy="false">)</mo>
 <mo>−</mo>
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>t</mi>
 <mo stretchy="false">)</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 </mstyle>
</mrow>
{\displaystyle w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971536adb2e247f013f2024b4aed421538935296" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:27.094ex; height:4.843ex;" alt="w_{f} (F) := \sup_{s, t \in F} | f(s) - f(t) |" /></dd>
</dl>

且对于<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>δ</mi>
 <mo>></mo>
 <mn>0</mn>
 </mstyle>
</mrow>
{\displaystyle \delta >0}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.343ex;" alt="\delta >0" />，将右连左极函数模（càdlàg modulus）定义为

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <msubsup>
 <mi>ϖ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 <mo>′</mo>
 </msubsup>
 <mo stretchy="false">(</mo>
 <mi>δ</mi>
 <mo stretchy="false">)</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">inf</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi mathvariant="normal">Π</mi>
 </mrow>
 </munder>
 <munder>
 <mo movablelimits="true" form="prefix">max</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mn>1</mn>
 <mo>≤</mo>
 <mi>i</mi>
 <mo>≤</mo>
 <mi>k</mi>
 </mrow>
 </munder>
 <msub>
 <mi>w</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 </msub>
 <mo stretchy="false">(</mo>
 <mo stretchy="false">[</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>i</mi>
 <mo>−</mo>
 <mn>1</mn>
 </mrow>
 </msub>
 <mo>,</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>i</mi>
 </mrow>
 </msub>
 <mo stretchy="false">)</mo>
 <mo stretchy="false">)</mo>
 <mo>,</mo>
 </mstyle>
</mrow>
{\displaystyle \varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d29f0cd91045936dc4a167d606a52d3b6d1dcc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:31.383ex; height:4.343ex;" alt="\varpi'_{f} (\delta) := \inf_{\Pi} \max_{1 \leq i \leq k} w_{f} ([t_{i - 1}, t_{i}))," /></dd>
</dl>

其中最大下界对所有划分<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi mathvariant="normal">Π</mi>
 <mo>=</mo>
 <mo fence="false" stretchy="false">{</mo>
 <mn>0</mn>
 <mo>=</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mn>0</mn>
 </mrow>
 </msub>
 <mo><</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mn>1</mn>
 </mrow>
 </msub>
 <mo><</mo>
 <mo>⋯</mo>
 <mo><</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>k</mi>
 </mrow>
 </msub>
 <mo>=</mo>
 <mi>T</mi>
 <mo fence="false" stretchy="false">}</mo>
 </mstyle>
</mrow>
{\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{k}=T\}}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fddbfea6ce116d064374fae84b3d5bfaf3fd18c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:34.076ex; height:2.843ex;" alt="\Pi = \{ 0 = t_{0} < t_{1} < \dots < t_{k} = T \}" />，<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>k</mi>
 <mo>∈</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi mathvariant="double-struck">N</mi>
 </mrow>
 </mstyle>
</mrow>
{\displaystyle k\in \mathbb {N} }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.761ex; height:2.176ex;" alt="k \in \mathbb{N}" />都存在，且<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <munder>
 <mo movablelimits="true" form="prefix">max</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>i</mi>
 </mrow>
 </munder>
 <mo stretchy="false">(</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>i</mi>
 </mrow>
 </msub>
 <mo>−</mo>
 <msub>
 <mi>t</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>i</mi>
 <mo>−</mo>
 <mn>1</mn>
 </mrow>
 </msub>
 <mo stretchy="false">)</mo>
 <mo><</mo>
 <mi>δ</mi>
 </mstyle>
</mrow>
{\displaystyle \max _{i}(t_{i}-t_{i-1})<\delta }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9626b2d03618bd162a55c0a7e2446b666c89421e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:18.636ex; height:4.009ex;" alt="\max_{i} (t_{i} - t_{i - 1}) < \delta" />。这一定义对于非右连左极函数<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 </mstyle>
</mrow>
{\displaystyle f}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.289ex; height:2.509ex;" alt="f" />是有意义的（就如通常的连续性模对于不连续函数是有意义的）且可以说明<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>f</mi>
 </mstyle>
</mrow>
{\displaystyle f}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.289ex; height:2.509ex;" alt="f" />是右连左极函数当且仅当<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>δ</mi>
 <mo stretchy="false">→</mo>
 <mn>0</mn>
 </mstyle>
</mrow>
{\displaystyle \delta \to 0}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2223650a165253bc7a1dcf49c7d41d42e2543350" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.857ex; height:2.343ex;" alt="\delta \to 0" />时<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <msubsup>
 <mi>ϖ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 <mo>′</mo>
 </msubsup>
 <mo stretchy="false">(</mo>
 <mi>δ</mi>
 <mo stretchy="false">)</mo>
 <mo stretchy="false">→</mo>
 <mn>0</mn>
 </mstyle>
</mrow>
{\displaystyle \varpi '_{f}(\delta )\to 0}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e9138e9aa1cdfc5ead5a1fb7996f51db55db00" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:10.765ex; height:3.343ex;" alt="\varpi'_{f} (\delta) \to 0" />。

这是令<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi mathvariant="normal">Λ</mi>
 </mstyle>
</mrow>
{\displaystyle \Lambda }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.623ex; height:2.343ex;" alt="\Lambda " />表示从<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>E</mi>
 </mstyle>
</mrow>
{\displaystyle E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.786ex; height:2.176ex;" alt="E" />到自身的所有严格递减的连续双射函数的集合（这些函数是“对时间的蠕动”）。令

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo fence="false" stretchy="false">∥</mo>
 <mi>f</mi>
 <mo>∥</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">sup</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>t</mi>
 <mo>∈</mo>
 <mi>E</mi>
 </mrow>
 </munder>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 <mi>f</mi>
 <mo stretchy="false">(</mo>
 <mi>t</mi>
 <mo stretchy="false">)</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 </mstyle>
</mrow>
{\displaystyle \|f\|:=\sup _{t\in E}|f(t)|}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21f45b3e8ea51a7a8d1aa3f28a937830309a1773" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:16.605ex; height:4.509ex;" alt="\| f \| := \sup_{t \in E} | f(t) |" /></dd>
</dl>

表示<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>E</mi>
 </mstyle>
</mrow>
{\displaystyle E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.786ex; height:2.176ex;" alt="E" />上的函数的一致范数。将<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>D</mi>
 </mstyle>
</mrow>
{\displaystyle D}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.935ex; height:2.176ex;" alt="D" /> 上的斯科罗霍德度量（Skorokhod metric）<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>σ</mi>
 </mstyle>
</mrow>
{\displaystyle \sigma }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.34ex; height:1.676ex;" alt="\sigma " />定义为

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>σ</mi>
 <mo stretchy="false">(</mo>
 <mi>f</mi>
 <mo>,</mo>
 <mi>g</mi>
 <mo stretchy="false">)</mo>
 <mo>:=</mo>
 <munder>
 <mo movablelimits="true" form="prefix">inf</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>λ</mi>
 <mo>∈</mo>
 <mi mathvariant="normal">Λ</mi>
 </mrow>
 </munder>
 <mo movablelimits="true" form="prefix">max</mo>
 <mo fence="false" stretchy="false">{</mo>
 <mo>∥</mo>
 <mi>λ</mi>
 <mo>−</mo>
 <mi>I</mi>
 <mo>∥</mo>
 <mo>,</mo>
 <mo>∥</mo>
 <mi>f</mi>
 <mo>−</mo>
 <mi>g</mi>
 <mo>∘</mo>
 <mi>λ</mi>
 <mo>∥</mo>
 <mo fence="false" stretchy="false">}</mo>
 <mo>,</mo>
 </mstyle>
</mrow>
{\displaystyle \sigma (f,g):=\inf _{\lambda \in \Lambda }\max\{\|\lambda -I\|,\|f-g\circ \lambda \|\},}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6a4d1f4bbfb14caa3484fe09fa17fac32c5d31" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:41.333ex; height:4.176ex;" alt="\sigma (f, g) := \inf_{\lambda \in \Lambda} \max \{ \| \lambda - I \|, \| f - g \circ \lambda \| \}," /></dd>
</dl>

其中<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>I</mi>
 <mo>:</mo>
 <mi>E</mi>
 <mo stretchy="false">→</mo>
 <mi>E</mi>
 </mstyle>
</mrow>
{\displaystyle I:E\to E}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2adceb8ece135eabd75d9dbb5a5b11b63767711f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.326ex; height:2.176ex;" alt="I : E \to E" />是恒等函数。以“蠕动”这种直观感觉来看，<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo fence="false" stretchy="false">∥</mo>
 <mi>λ</mi>
 <mo>−</mo>
 <mi>I</mi>
 <mo fence="false" stretchy="false">∥</mo>
 </mstyle>
</mrow>
{\displaystyle \|\lambda -I\|}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e411b673d9d96ff05f35e906c8ff079bae9abbb9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.745ex; height:2.843ex;" alt="\| \lambda - I \|" />度量了“时间的蠕动”，而<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo fence="false" stretchy="false">∥</mo>
 <mi>f</mi>
 <mo>−</mo>
 <mi>g</mi>
 <mo>∘</mo>
 <mi>λ</mi>
 <mo fence="false" stretchy="false">∥</mo>
 </mstyle>
</mrow>
{\displaystyle \|f-g\circ \lambda \|}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69aed72b497be85771d72d19f503cfd59ead871" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:11.183ex; height:2.843ex;" alt="\| f - g \circ \lambda \|" />度量了“空间的蠕动”。

我们可以证明斯科罗霍德度量度量的确是度量。由<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>σ</mi>
 </mstyle>
</mrow>
{\displaystyle \sigma }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.34ex; height:1.676ex;" alt="\sigma " />生成的拓扑<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi mathvariant="normal">Σ</mi>
 </mstyle>
</mrow>
{\displaystyle \Sigma }</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.689ex; height:2.176ex;" alt="\Sigma " />称为<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mi>D</mi>
 </mstyle>
</mrow>
{\displaystyle D}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.935ex; height:2.176ex;" alt="D" />上的斯科罗霍德拓扑（Skorokhod topology）。

[####]

[#####]

虽然D 不是关于斯科罗霍德度量σ 的一个完备空间，但是可以证明存在具完备性的关于D 的拓扑等价度量 σ0 。

[######]

关于σ 或σ0 的D 是可分空间，因此斯科罗霍德空间是Polish空间。

[#######]

通过应用阿尔泽拉－阿斯科利定理，我们可以证明斯科罗霍德空间D 上概率测度的一个序列<math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo stretchy="false">(</mo>
 <msub>
 <mi>μ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 </mrow>
 </msub>
 <msubsup>
 <mo stretchy="false">)</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 <mo>=</mo>
 <mn>1</mn>
 </mrow>
 <mrow class="MJX-TeXAtom-ORD">
 <mi mathvariant="normal">∞</mi>
 </mrow>
 </msubsup>
 </mstyle>
</mrow>
{\displaystyle (\mu _{n})_{n=1}^{\infty }}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33a74fab09d45838666e1e06963ed7218acfefdb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:7.809ex; height:3.009ex;" alt="(\mu_{n})_{n = 1}^{\infty}" />是胎紧的当且仅当同时满足下列两个条件：

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <munder>
 <mo movablelimits="true" form="prefix">lim</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>a</mi>
 <mo stretchy="false">→</mo>
 <mi mathvariant="normal">∞</mi>
 </mrow>
 </munder>
 <munder>
 <mo movablelimits="true" form="prefix">lim sup</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 <mo stretchy="false">→</mo>
 <mi mathvariant="normal">∞</mi>
 </mrow>
 </munder>
 <msub>
 <mi>μ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 </mrow>
 </msub>
 <mo fence="false" stretchy="false">{</mo>
 <mi>f</mi>
 <mo>∈</mo>
 <mi>D</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 <mo>∥</mo>
 <mi>f</mi>
 <mo>∥</mo>
 <mo>≥</mo>
 <mi>a</mi>
 <mo fence="false" stretchy="false">}</mo>
 <mo>=</mo>
 <mn>0</mn>
 <mo>,</mo>
 </mstyle>
</mrow>
{\displaystyle \lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}\{f\in D|\|f\|\geq a\}=0,}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24c5a60efab478dee34eb0f3903d07a2f4b3faa5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:36.773ex; height:4.343ex;" alt="\lim_{a \to \infty} \limsup_{n \to \infty} \mu_{n} \{ f \in D | \| f \| \geq a \} = 0," /></dd>
</dl>

和

<dl>
<dd><math xmlns="http://www.w3.org/1998/Math/MathML" >
 <semantics>
<mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <munder>
 <mo movablelimits="true" form="prefix">lim</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>δ</mi>
 <mo stretchy="false">→</mo>
 <mn>0</mn>
 </mrow>
 </munder>
 <munder>
 <mo movablelimits="true" form="prefix">lim sup</mo>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 <mo stretchy="false">→</mo>
 <mi mathvariant="normal">∞</mi>
 </mrow>
 </munder>
 <msub>
 <mi>μ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>n</mi>
 </mrow>
 </msub>
 <mo fence="false" stretchy="false">{</mo>
 <mi>f</mi>
 <mo>∈</mo>
 <mi>D</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mo stretchy="false">|</mo>
 </mrow>
 <msubsup>
 <mi>ϖ</mi>
 <mrow class="MJX-TeXAtom-ORD">
 <mi>f</mi>
 </mrow>
 <mo>′</mo>
 </msubsup>
 <mo stretchy="false">(</mo>
 <mi>δ</mi>
 <mo stretchy="false">)</mo>
 <mo>≥</mo>
 <mi>ε</mi>
 <mo fence="false" stretchy="false">}</mo>
 <mo>=</mo>
 <mn>0</mn>
 <mrow class="MJX-TeXAtom-ORD">
 <mtext> for all </mtext>
 </mrow>
 <mi>ε</mi>
 <mo>></mo>
 <mn>0.</mn>
 </mstyle>
</mrow>
{\displaystyle \lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}\{f\in D|\varpi '_{f}(\delta )\geq \varepsilon \}=0{\text{ for all }}\varepsilon >0.}</annotation>
 </semantics>
</math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b85777f2b90be01d5a5ee4cad07efc29967816b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:50.465ex; height:4.343ex;" alt="\lim_{\delta \to 0} \limsup_{n \to \infty} \mu_{n} \{ f \in D | \varpi'_{f} (\delta) \geq \varepsilon \} = 0\text{ for all }\varepsilon > 0." /></dd>
</dl>
[########]

在斯科罗霍德拓扑和函数的逐点加法下，D 不是一个拓扑群。

[#########]

<ul>
<li><cite class="citation book">Billingsley, Patrick. Probability and Measure. New York, NY: John Wiley & Sons, Inc. 1995. ISBN 0-471-00710-2.</cite> </li>
<li><cite class="citation book">Billingsley, Patrick. Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. 1999. ISBN 0-471-19745-9.</cite> </li>
</ul>

分类：<ul><li>实分析</li><li>随机过程</li><li>函数</li></ul>
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