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<title>Random element</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random element</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Probability_theory" title="Probability theory">probability theory</a>, <b>random element</b> is a generalization of the concept of <a href="Random_variable" title="Random variable">random variable</a> to more complicated spaces than the simple real line. The concept was introduced by <a href="Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> (<a href="#CITEREFFréchet1948">1948</a>) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, <a href="Random_vector" class="mw-redirect" title="Random vector">vectors</a>, <a href="Function_(mathematics)" title="Function (mathematics)">functions</a>, processes, <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>, <a href="Series_(mathematics)" title="Series (mathematics)">series</a>, <a href="Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">transformations</a>, and also <a href="Set_(mathematics)" title="Set (mathematics)">sets</a> or collections of sets.”<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The modern-day usage of “random element” frequently assumes the space of values is a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a>, often a <a href="Banach_space" title="Banach space">Banach</a> or <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> with a specified natural <a href="Sigma_algebra" class="mw-redirect" title="Sigma algebra">sigma algebra</a> of subsets.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

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</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> be a <a href="Probability_space" title="Probability space">probability space</a>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})}">

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<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})}</annotation>

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</math></span><img src="./497b309dc3f6f358722b8a00936c8e1ed5c787b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.93ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})}" loading="lazy"></span> a <a href="Measurable_space" title="Measurable space">measurable space</a>. A <b>random element</b> with values in <i>E</i> is a function <span class="nowrap"><i>X</i>: Ω→<i>E</i></span> which is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {F}},{\mathcal {E}})}">

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<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {F}},{\mathcal {E}})}</annotation>

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</math></span><img src="./13ce06551407215a8f9b81d6eeb44522b29302d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {F}},{\mathcal {E}})}" loading="lazy"></span>-<a href="Measurable_function" title="Measurable function">measurable</a>. That is, a function X such that for any <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in {\mathcal {E}}}">

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<annotation encoding="application/x-tex">{\displaystyle B\in {\mathcal {E}}}</annotation>

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</math></span><img src="./81710349199c3fc6bcbe14f6758e0c2f1fe9bbca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle B\in {\mathcal {E}}}" loading="lazy"></span>, the <a href="Preimage" class="mw-redirect" title="Preimage">preimage</a> of B lies in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>.

</p><p>Sometimes random elements with values in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span> are called <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math></span><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span>-valued random variables.

</p><p>Note if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}">

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<annotation encoding="application/x-tex">{\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}</annotation>

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</math></span><img src="./8ebde5e32b515628edba05c31377413ea4edfe13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.58ex; height:2.843ex;" alt="{\displaystyle (E,{\mathcal {E}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> are the real numbers, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}(\mathbb {R} )}</annotation>

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</math></span><img src="./b72c5154b8532f1f97a9d217a1ec867e934e772f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.031ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}(\mathbb {R} )}" loading="lazy"></span> is its <a href="Borel_%CF%83-algebra" class="mw-redirect" title="Borel σ-algebra">Borel σ-algebra</a>, then the definition of random element is the classical definition of <a href="Random_variable" title="Random variable">random variable</a>.

</p><p>The definition of a random element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> with values in a <a href="Banach_space" title="Banach space">Banach space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is typically understood to utilize the smallest <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span>-algebra on <i>B</i> for which every <a href="Bounded_linear_functional" class="mw-redirect" title="Bounded linear functional">bounded linear functional</a> is measurable. An equivalent definition, in this case, to the above, is that a map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\Omega \rightarrow B}">

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<annotation encoding="application/x-tex">{\displaystyle X:\Omega \rightarrow B}</annotation>

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</math></span><img src="./5b5235532c808fd3492773ebf634d54e86074afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.973ex; height:2.176ex;" alt="{\displaystyle X:\Omega \rightarrow B}" loading="lazy"></span>, from a probability space, is a random element if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ X}">

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<annotation encoding="application/x-tex">{\displaystyle f\circ X}</annotation>

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</math></span><img src="./631a13533ca4d6ce25d3698a87199833ae03a8cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.453ex; height:2.509ex;" alt="{\displaystyle f\circ X}" loading="lazy"></span> is a random variable for every bounded linear functional <i>f</i>, or, equivalently, that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is <a href="Weakly_measurable_function" title="Weakly measurable function">weakly measurable</a>.

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_random_elements">Examples of random elements</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Random_variable">Random variable</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_variable" title="Random variable">Random variable</a></div>

<p>A <i>random variable</i> is the simplest type of random element. It is a map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\colon \Omega \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle X\colon \Omega \to \mathbb {R} }</annotation>

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</math></span><img src="./b68ecb50490003c72b7d6627145965f602b302b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.984ex; height:2.176ex;" alt="{\displaystyle X\colon \Omega \to \mathbb {R} }" loading="lazy"></span> is a <a href="Measurable_function" title="Measurable function">measurable function</a> from the set of possible outcomes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>.

</p><p>As a real-valued function, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> often describes some numerical quantity of a given event. E.g. the number of heads after a certain number of coin flips; the heights of different people.

</p><p>When the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> (or range) of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is finite or <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a>, the random variable is called a discrete random variable<sup id="cite_ref-Yates_3-0" class="reference"><a href="#cite_note-Yates-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and its distribution can be described by a <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> which assigns a probability to each value in the image of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. If the image is uncountably infinite then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is called a continuous random variable. In the special case that it is <a href="Absolutely_continuous" class="mw-redirect" title="Absolutely continuous">absolutely continuous</a>, its distribution can be described by a <a href="Probability_density_function" title="Probability density function">probability density function</a>, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for example a <a href="Mixture_distribution" title="Mixture distribution">mixture distribution</a>. Such random variables cannot be described by a probability density or a probability mass function.

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<div class="mw-heading mw-heading3"><h3 id="Random_vector">Random vector</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_vector" class="mw-redirect" title="Random vector">Random vector</a></div>

<p>A <b>random vector</b> is a <a href="Column_vector" class="mw-redirect" title="Column vector">column</a> <a href="Vector_space" title="Vector space">vector</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}</annotation>

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</math></span><img src="./bc319ec653bf15480e36b48c0d4029b8244dfbce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.607ex; height:3.176ex;" alt="{\displaystyle \mathbf {X} =(X_{1},...,X_{n})^{T}}" loading="lazy"></span> (or its <a href="Transpose" title="Transpose">transpose</a>, which is a <a href="Row_vector" class="mw-redirect" title="Row vector">row vector</a>) whose components are <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>-valued <a href="Random_variable" title="Random variable">random variables</a> on the same <a href="Probability_space" title="Probability space">probability space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

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<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

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</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math></span><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"></span> is the <a href="Sample_space" title="Sample space">sample space</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span> is the <a href="Sigma-algebra" class="mw-redirect" title="Sigma-algebra">sigma-algebra</a> (the collection of all events), and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is the <a href="Probability_measure" title="Probability measure">probability measure</a> (a function returning each event's <a href="Probability" title="Probability">probability</a>).

</p><p>Random vectors are often used as the underlying implementation of various types of aggregate <a href="Random_variable" title="Random variable">random variables</a>, e.g. a <a href="Random_matrix" title="Random matrix">random matrix</a>, <a href="Random_tree" title="Random tree">random tree</a>, <a href="Random_sequence" title="Random sequence">random sequence</a>, <a href="Random_process" class="mw-redirect" title="Random process">random process</a>, etc.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_matrix">Random matrix</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_matrix_theory" class="mw-redirect" title="Random matrix theory">Random matrix theory</a></div>

<p>A <b>random matrix</b> is a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>-valued random element. Many important properties of <a href="Physical_system" title="Physical system">physical systems</a> can be represented mathematically as matrix problems. For example, the <a href="Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> of a <a href="Lattice_model_(physics)" title="Lattice model (physics)">lattice</a> can be computed from the dynamical matrix of the particle-particle interactions within the lattice.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_function">Random function</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_function" class="mw-redirect" title="Random function">Random function</a></div>

<p>A random function is a type of random element in which a single outcome is selected from some family of functions, where the family consists some class of all maps from the <a href="Domain_of_a_function" title="Domain of a function">domain</a> to the <a href="Codomain" title="Codomain">codomain</a>. For example, the class may be restricted to all <a href="Continuous_function" title="Continuous function">continuous functions</a> or to all <a href="Step_function" title="Step function">step functions</a>. The values determined by a random function evaluated at different points from the same realization would not generally be <a href="Statistically_independent" class="mw-redirect" title="Statistically independent">statistically independent</a> but, depending on the model, values determined at the same or different points from different realisations might well be treated as independent.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_process">Random process</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_process" class="mw-redirect" title="Random process">Random process</a></div>

<p>A <b>Random process</b> is a collection of <a href="Random_variable" title="Random variable">random variables</a>, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or <a href="Deterministic_system" title="Deterministic system">deterministic system</a>). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

</p><p>In the simple case of <a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">discrete time</a>, as opposed to <a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">continuous time</a>, a stochastic process involves a <a href="Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a> of random variables and the <a href="Time_series" title="Time series">time series</a> associated with these random variables (for example, see <a href="Markov_chain" title="Markov chain">Markov chain</a>, also known as discrete-time Markov chain).

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_field">Random field</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_field" title="Random field">Random field</a></div>

<p>Given a <a href="Probability_space" title="Probability space">probability space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},P)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},P)}</annotation>

</semantics>

</math></span><img src="./9d77104a5c3c49cc0634dcf6908db7ad45f738d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.227ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},P)}" loading="lazy"></span> and a <a href="Measurable_space" title="Measurable space">measurable space</a> X,

an <i>X</i>-valued random field is a collection of <i>X</i>-valued

<a href="Random_variable" title="Random variable">random variables</a> indexed by elements in a topological space <i>T</i>. That is, a random field <i>F</i> is a collection

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{F_{t}:t\in T\}}</annotation>

</semantics>

</math></span><img src="./94db7b070979164f8b2d60eae421b869bda557fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.899ex; height:2.843ex;" alt="{\displaystyle \{F_{t}:t\in T\}}" loading="lazy"></span></dd></dl>

<p>where each <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{t}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{t}}</annotation>

</semantics>

</math></span><img src="./42cf9007bfa40c26216b648363be8f67e73d38cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.321ex; height:2.509ex;" alt="{\displaystyle F_{t}}" loading="lazy"></span> is an <i>X</i>-valued random variable.

</p><p>Several kinds of random fields exist, among them the <a href="Markov_random_field" title="Markov random field">Markov random field</a> (MRF), <a href="Gibbs_random_field" class="mw-redirect" title="Gibbs random field">Gibbs random field</a> (GRF), <a href="Conditional_random_field" title="Conditional random field">conditional random field</a> (CRF), and <a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a>. An MRF exhibits the Markovian property

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|\partial _{i}),\,}</annotation>

</semantics>

</math></span><img src="./8875edfb889ffc590be16f53fd398cde2c55d57d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.538ex; height:3.009ex;" alt="{\displaystyle P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|\partial _{i}),\,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{i}}</annotation>

</semantics>

</math></span><img src="./9b6be63a3ddb914c608027404afea96933df7c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.509ex;" alt="{\displaystyle \partial _{i}}" loading="lazy"></span> is a set of neighbours of the random variable <i>X</i><sub><i>i</i></sub>. In other words, the probability that a random variable assumes a value depends on the other random variables only through the ones that are its immediate neighbours. The probability of a random variable in an MRF is given by

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</munder>

<msup>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(X_{i}=x_{i}|\partial _{i})={\frac {P(\omega )}{\sum _{\omega '}P(\omega ')}},}</annotation>

</semantics>

</math></span><img src="./6a0689dcdeee880aa0a87d85348a2aade2a94646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.081ex; height:6.509ex;" alt="{\displaystyle P(X_{i}=x_{i}|\partial _{i})={\frac {P(\omega )}{\sum _{\omega '}P(\omega ')}},}" loading="lazy"></span></dd></dl>

<p>where Ω' is the same realization of Ω, except for random variable <i>X</i><sub><i>i</i></sub>. It is difficult to calculate with this equation, without recourse to the relation between MRFs and GRFs proposed by <a href="Julian_Besag" title="Julian Besag">Julian Besag</a> in 1974.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_measure">Random measure</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_measure" title="Random measure">Random measure</a></div>

<p>A <b>random measure</b> is a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>-valued random element.<sup id="cite_ref-RN_5-0" class="reference"><a href="#cite_note-RN-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-G_6-0" class="reference"><a href="#cite_note-G-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Let X be a complete separable metric space and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}(X)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}(X)}</annotation>

</semantics>

</math></span><img src="./43fdb56b20a6cdeccd661b706552f756e235fb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.844ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}(X)}" loading="lazy"></span> the <a href="Sigma-algebra" class="mw-redirect" title="Sigma-algebra">σ-algebra</a> of its Borel sets. A <a href="Borel_measure" title="Borel measure">Borel measure</a> μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}}</annotation>

</semantics>

</math></span><img src="./a6781b549988e883088f5c5994e9011936e2d02f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.886ex; height:2.509ex;" alt="{\displaystyle M_{X}}" loading="lazy"></span> be the space of all boundedly finite measures on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}(X)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}(X)}</annotation>

</semantics>

</math></span><img src="./43fdb56b20a6cdeccd661b706552f756e235fb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.844ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}(X)}" loading="lazy"></span>. Let <span class="nowrap">(Ω, ℱ, <i>P</i>)</span> be a <a href="Probability_space" title="Probability space">probability space</a>, then a random measure maps from this probability space to the <a href="Measurable_space" title="Measurable space">measurable space</a> <span class="nowrap">(<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{X}}</annotation>

</semantics>

</math></span><img src="./a6781b549988e883088f5c5994e9011936e2d02f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.886ex; height:2.509ex;" alt="{\displaystyle M_{X}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {B}}(M_{X})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {B}}(M_{X})}</annotation>

</semantics>

</math></span><img src="./63a82a5e46953db030ae3df2aa9ae47ffbb95f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.75ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {B}}(M_{X})}" loading="lazy"></span>)</span>.<sup id="cite_ref-daleyPPI2003_7-0" class="reference"><a href="#cite_note-daleyPPI2003-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> A measure generally might be decomposed as:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{d}+\mu _{a}=\mu _{d}+\sum _{n=1}^{N}\kappa _{n}\delta _{X_{n}},}</annotation>

</semantics>

</math></span><img src="./3a92517d752acc4f91b17713fb056042ab3a9a59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.307ex; height:7.343ex;" alt="{\displaystyle \mu =\mu _{d}+\mu _{a}=\mu _{d}+\sum _{n=1}^{N}\kappa _{n}\delta _{X_{n}},}" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{d}}</annotation>

</semantics>

</math></span><img src="./1f5a79bfd0a50678913082f322d849349359975c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle \mu _{d}}" loading="lazy"></span> is a diffuse measure without atoms, while <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu _{a}}</annotation>

</semantics>

</math></span><img src="./ea3dc3301612e553675341c662acee2c98f8607f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.504ex; height:2.176ex;" alt="{\displaystyle \mu _{a}}" loading="lazy"></span> is a purely atomic measure.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_set">Random set</h3></div>

<p>A random set is a set-valued random element.

</p><p>One specific example is a <a href="Random_compact_set" title="Random compact set">random compact set</a>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,d)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (M,d)}</annotation>

</semantics>

</math></span><img src="./d78e6f2ddf5baee227ee2a9f164726ba0c23c263.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}" loading="lazy"></span> be a <a href="Complete_space" class="mw-redirect" title="Complete space">complete</a> <a href="Separable_space" title="Separable space">separable</a> <a href="Metric_space" title="Metric space">metric space</a>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

</semantics>

</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span> denote the set of all compact subsets of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>. The Hausdorff metric <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math></span><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

</semantics>

</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span> is defined by

</p>

<msub>

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</msub>

<msub>

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<mrow>

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</msub>

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</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}</annotation>

</semantics>

</math></span><img src="./31cb6e7540ed1218a3cb7557b1273a2365fd1bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.072ex; height:7.509ex;" alt="{\displaystyle h(K_{1},K_{2}):=\max \left\{\sup _{a\in K_{1}}\inf _{b\in K_{2}}d(a,b),\sup _{b\in K_{2}}\inf _{a\in K_{1}}d(a,b)\right\}.}" loading="lazy"></span></dd></dl>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {K}},h)}</annotation>

</semantics>

</math></span><img src="./77799a9013e23eb7fa77f49eadd7b1272d8e89a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.953ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {K}},h)}" loading="lazy"></span> is also а complete separable metric space. The corresponding open subsets generate a <a href="Sigma_algebra" class="mw-redirect" title="Sigma algebra">σ-algebra</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

</semantics>

</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span>, the <a href="Borel_sigma_algebra" class="mw-redirect" title="Borel sigma algebra">Borel sigma algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {B}}({\mathcal {K}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {B}}({\mathcal {K}})}</annotation>

</semantics>

</math></span><img src="./f380c186f6df6283689998013e31c15306fd62be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.124ex; height:2.843ex;" alt="{\displaystyle {\mathcal {B}}({\mathcal {K}})}" loading="lazy"></span> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {K}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {K}}}</annotation>

</semantics>

</math></span><img src="./3a70fc5d5ef4fa8ce694447bef39c1aa167a68b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.176ex;" alt="{\displaystyle {\mathcal {K}}}" loading="lazy"></span>.

</p><p>A <b>random compact set</b> is а <a href="Measurable_function" title="Measurable function">measurable function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> from а <a href="Probability_space" title="Probability space">probability space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}</annotation>

</semantics>

</math></span><img src="./6bb8743f7565082ed1a9ee0490d9d71be82eafaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.902ex; height:2.843ex;" alt="{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}" loading="lazy"></span> into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}</annotation>

</semantics>

</math></span><img src="./927d1a33c8d33e124a53c25b1397cd0c19df91bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.738ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {K}},{\mathcal {B}}({\mathcal {K}}))}" loading="lazy"></span>.

</p><p>Put another way, a random compact set is a measurable function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\colon \Omega \to 2^{M}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K\colon \Omega \to 2^{M}}</annotation>

</semantics>

</math></span><img src="./b30b1c57552c73f5f747e054dd35eb069608bf6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.514ex; height:2.676ex;" alt="{\displaystyle K\colon \Omega \to 2^{M}}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\omega )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\omega )}</annotation>

</semantics>

</math></span><img src="./21bef70b39ac777e9fe32525939280902d41b56d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.321ex; height:2.843ex;" alt="{\displaystyle K(\omega )}" loading="lazy"></span> is <a href="Almost_surely" title="Almost surely">almost surely</a> compact and

</p>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}</annotation>

</semantics>

</math></span><img src="./658f5b388674d664aaecea6fc8d58a884c225127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.398ex; height:4.509ex;" alt="{\displaystyle \omega \mapsto \inf _{b\in K(\omega )}d(x,b)}" loading="lazy"></span></dd></dl>

<p>is a measurable function for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in M}</annotation>

</semantics>

</math></span><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Random_geometric_objects">Random geometric objects</h3></div>

<p>These include random points, random figures,<sup id="cite_ref-Stoyan_8-0" class="reference"><a href="#cite_note-Stoyan-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> and random shapes.<sup id="cite_ref-Stoyan_8-1" class="reference"><a href="#cite_note-Stoyan-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFFréchet1948" class="citation journal cs1"><a href="Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Fréchet, M.</a> (1948). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=AIHP_1948__10_4_215_0">"Les éléments aléatoires de nature quelconque dans un espace distancié"</a>. <i><a href="Annales_de_l'Institut_Henri_Poincar%C3%A9" class="mw-redirect" title="Annales de l'Institut Henri Poincaré">Annales de l'Institut Henri Poincaré</a></i>. <b>10</b> (4): <span class="nowrap">215–</span>310.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">V.V. Buldygin, A.B. Kharazishvili. Geometric Aspects of Probability Theory and Mathematical Statistics. – Kluwer Academic Publishers, Dordrecht. – 2000</span>

</li>

<li id="cite_note-Yates-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yates_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFYatesMooreStarnes2003" class="citation book cs1">Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/"><i>The Practice of Statistics</i></a> (2nd ed.). New York: <a href="W._H._Freeman_and_Company" title="W. H. Freeman and Company">Freeman</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7167-4773-4</bdi>. Archived from <a rel="nofollow" class="external text" href="http://bcs.whfreeman.com/yates2e/">the original</a> on 2005-02-09.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFL._CastañedaV._ArunachalamS._Dharmaraja2012" class="citation book cs1">L. Castañeda; V. Arunachalam & S. Dharmaraja (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zxXRn-Qmtk8C&pg=PA67"><i>Introduction to Probability and Stochastic Processes with Applications</i></a>. Wiley. p. 67. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118344941</bdi>.</cite></span>

</li>

<li id="cite_note-RN-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-RN_5-0">^</a></b></span> <span class="reference-text"><a href="Olav_Kallenberg" title="Olav Kallenberg">Kallenberg, O.</a>, <i>Random Measures</i>, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin (1986). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-394960-2</bdi> <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0854102">0854102</a>. An authoritative but rather difficult reference.</span>

</li>

<li id="cite_note-G-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-G_6-0">^</a></b></span> <span class="reference-text">Jan Grandell, Point processes and random measures, <i>Advances in Applied Probability</i> 9 (1977) 502-526. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0478331">0478331</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1426111">JSTOR</a> A nice and clear introduction.</span>

</li>

<li id="cite_note-daleyPPI2003-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-daleyPPI2003_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDaleyVere-Jones2003" class="citation book cs1">Daley, D. J.; Vere-Jones, D. (2003). <i>An Introduction to the Theory of Point Processes</i>. Probability and its Applications. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fb97277">10.1007/b97277</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95541-0</bdi>.</cite></span>

</li>

<li id="cite_note-Stoyan-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Stoyan_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Stoyan_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Stoyan, D., and Stoyan, H. (1994) <i>Fractals, Random Shapes and Point Fields. Methods of Geometrical Statistics</i>. Chichester, New York: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-93757-6</bdi></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Literature">Literature</h3></div>

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<ul><li>Hoffman-Jorgensen J., Pisier G. (1976) "Ann.Probab.", v.4, 587–589.</li>

<li>Mourier E. (1955) Elements aleatoires dans un espace de Banach (These). Paris.</li>

<li>Prokhorov Yu.V. (1999) Random element. Probability and Mathematical statistics. Encyclopedia. Moscow: "Great Russian Encyclopedia", P.623.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Random_element">Entry in Springer Encyclopedia of Mathematics</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-10-13" href="https://en.wikipedia.org/wiki/?title=Random_element&oldid=1179987304">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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