SherbyNet - /page/wikibook/entry.mu

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Rate function</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a> — specifically, in <a href="Large_deviations_theory" title="Large deviations theory">large deviations theory</a> — a <b>rate function</b> is a function used to quantify the <a href="Probability" title="Probability">probabilities</a> of rare events. Such functions are used to formulate <b>large deviation principles</b>. A large deviation principle quantifies the asymptotic probability of rare events for a sequence of probabilities.

</p><p>A <b>rate function</b> is also called a <b>Cramér function</b>, after the Swedish probabilist <a href="Harald_Cram%C3%A9r" title="Harald Cramér">Harald Cramér</a>.

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

<p><b>Rate function </b>An <a href="Extended_real_number_line" title="Extended real number line">extended real-valued</a> 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 I:X\to [0,+\infty ]}">

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<annotation encoding="application/x-tex">{\displaystyle I:X\to [0,+\infty ]}</annotation>

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</math></span><img src="./4e7fb7c0ce3895f5980533c97f8d77661aea4293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.325ex; height:2.843ex;" alt="{\displaystyle I:X\to [0,+\infty ]}" loading="lazy"></span> defined on a <a href="Hausdorff_space" title="Hausdorff space">Hausdorff</a> <a href="Topological_space" title="Topological space">topological 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 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 said to be a <b>rate function</b> if it is not identically <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 +\infty }">

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

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</math></span><img src="./bddbb0e4420a7e744cf71bd71216e11b0bf88831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }" loading="lazy"></span> and is <a href="Lower_semi-continuous" class="mw-redirect" title="Lower semi-continuous">lower semi-continuous</a> <i>i.e.</i> all the sub-level sets

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<annotation encoding="application/x-tex">{\displaystyle \{x\in X\mid I(x)\leq c\}{\mbox{ for }}c\geq 0}</annotation>

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</math></span><img src="./509f875ad5117f455c34c447f39297f155c8c462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.043ex; height:2.843ex;" alt="{\displaystyle \{x\in X\mid I(x)\leq c\}{\mbox{ for }}c\geq 0}" loading="lazy"></span></dd></dl>

<p>are <a href="Closed_set" title="Closed set">closed</a> 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 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, furthermore, they are <a href="Compact_space" title="Compact space">compact</a>, 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 I}">

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

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</math></span><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> is said to be a <b>good rate function</b>.

</p><p>A family of <a href="Probability_measure" title="Probability measure">probability measures</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 (\mu _{\delta })_{\delta >0}}">

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<annotation encoding="application/x-tex">{\displaystyle (\mu _{\delta })_{\delta >0}}</annotation>

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</math></span><img src="./a71b0d98074b340e80739e10f50148cabda9ac0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:2.843ex;" alt="{\displaystyle (\mu _{\delta })_{\delta >0}}" 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 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 said to satisfy the <b>large deviation principle</b> with rate 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 I:X\to [0,+\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle I:X\to [0,+\infty )}</annotation>

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</math></span><img src="./f05af5dfb6a9711449033ed781a7e5c23e9a84dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.583ex; height:2.843ex;" alt="{\displaystyle I:X\to [0,+\infty )}" loading="lazy"></span> (and rate <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 1/\delta }">

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<annotation encoding="application/x-tex">{\displaystyle 1/\delta }</annotation>

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</math></span><img src="./6c5382fc7faa85b5bbd8e06331aa061c3daa9b91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.374ex; height:2.843ex;" alt="{\displaystyle 1/\delta }" loading="lazy"></span>) if, for every closed set <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\subseteq X}">

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

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</math></span><img src="./17dbca630c46b25edb7c7113d35770b047bbc476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.819ex; height:2.343ex;" alt="{\displaystyle F\subseteq X}" loading="lazy"></span> and every <a href="Open_set" title="Open set">open set</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 G\subseteq X}">

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

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</math></span><img src="./906492ca71b34c4259c753e9b26f386d0d2e3fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.905ex; height:2.343ex;" alt="{\displaystyle G\subseteq X}" loading="lazy"></span>,

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(F)\leq -\inf _{x\in F}I(x),\quad {\mbox{(U)}}}</annotation>

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</math></span><img src="./bd5d78c008a6f186917193a58e15473133e87d78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.394ex; height:4.843ex;" alt="{\displaystyle \limsup _{\delta \downarrow 0}\delta \log \mu _{\delta }(F)\leq -\inf _{x\in F}I(x),\quad {\mbox{(U)}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \liminf _{\delta \downarrow 0}\delta \log \mu _{\delta }(G)\geq -\inf _{x\in G}I(x).\quad {\mbox{(L)}}}</annotation>

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</math></span><img src="./b973bf8cb9d32a58424a7e841bd71ce3e091ea09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.4ex; height:4.343ex;" alt="{\displaystyle \liminf _{\delta \downarrow 0}\delta \log \mu _{\delta }(G)\geq -\inf _{x\in G}I(x).\quad {\mbox{(L)}}}" loading="lazy"></span></dd></dl>

<p>If the upper bound (U) holds only for compact (instead of closed) sets <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}">

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

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</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span>, 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 (\mu _{\delta })_{\delta >0}}">

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<annotation encoding="application/x-tex">{\displaystyle (\mu _{\delta })_{\delta >0}}</annotation>

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</math></span><img src="./a71b0d98074b340e80739e10f50148cabda9ac0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:2.843ex;" alt="{\displaystyle (\mu _{\delta })_{\delta >0}}" loading="lazy"></span> is said to satisfy the <b>weak large deviations principle</b> (with rate <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 1/\delta }">

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<annotation encoding="application/x-tex">{\displaystyle 1/\delta }</annotation>

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</math></span><img src="./6c5382fc7faa85b5bbd8e06331aa061c3daa9b91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.374ex; height:2.843ex;" alt="{\displaystyle 1/\delta }" loading="lazy"></span> and weak rate 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 I}">

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

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</math></span><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span>).

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

<p>The role of the open and closed sets in the large deviation principle is similar to their role in the weak convergence of probability measures: recall 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 (\mu _{\delta })_{\delta >0}}">

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<annotation encoding="application/x-tex">{\displaystyle (\mu _{\delta })_{\delta >0}}</annotation>

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</math></span><img src="./a71b0d98074b340e80739e10f50148cabda9ac0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:2.843ex;" alt="{\displaystyle (\mu _{\delta })_{\delta >0}}" loading="lazy"></span> is said to converge weakly 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 \mu }">

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

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</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> if, for every closed set <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\subseteq X}">

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</math></span><img src="./17dbca630c46b25edb7c7113d35770b047bbc476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.819ex; height:2.343ex;" alt="{\displaystyle F\subseteq X}" loading="lazy"></span> and every <a href="Open_set" title="Open set">open set</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 G\subseteq X}">

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</math></span><img src="./906492ca71b34c4259c753e9b26f386d0d2e3fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.905ex; height:2.343ex;" alt="{\displaystyle G\subseteq X}" loading="lazy"></span>,

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{\delta \downarrow 0}\mu _{\delta }(F)\leq \mu (F),}</annotation>

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</math></span><img src="./bf82f9bd0b56966fedde53a1e3e08286b1b606f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.128ex; height:4.843ex;" alt="{\displaystyle \limsup _{\delta \downarrow 0}\mu _{\delta }(F)\leq \mu (F),}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle \liminf _{\delta \downarrow 0}\mu _{\delta }(G)\geq \mu (G).}</annotation>

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</math></span><img src="./c979462f739825c959bbe3faab2c0442447c03b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.45ex; height:4.343ex;" alt="{\displaystyle \liminf _{\delta \downarrow 0}\mu _{\delta }(G)\geq \mu (G).}" loading="lazy"></span></dd></dl>

<p>There is some variation in the nomenclature used in the literature: for example, den Hollander (2000) uses simply "rate function" where this article — following Dembo & Zeitouni (1998) — uses "good rate function", and "weak rate function". Rassoul-Agha & Seppäläinen (2015) uses the term "tight rate function" instead of "good rate function" due to the connection with exponential tightness of a family of measures. Regardless of the nomenclature used for rate functions, examination of whether the upper bound inequality (U) is supposed to hold for closed or compact sets tells one whether the large deviation principle in use is strong or weak.

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

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

<p>A natural question to ask, given the somewhat abstract setting of the general framework above, is whether the rate function is unique. This turns out to be the case: given a sequence of probability measures (<i>μ</i><sub><i>δ</i></sub>)<sub><i>δ</i>>0</sub> on <i>X</i> satisfying the large deviation principle for two rate functions <i>I</i> and <i>J</i>, it follows that <i>I</i>(<i>x</i>) = <i>J</i>(<i>x</i>) for all <i>x</i> ∈ <i>X</i>.

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<div class="mw-heading mw-heading3"><h3 id="Exponential_tightness">Exponential tightness</h3></div>

<p>It is possible to convert a weak large deviation principle into a strong one if the measures converge sufficiently quickly. If the upper bound holds for compact sets <i>F</i> and the sequence of measures (<i>μ</i><sub><i>δ</i></sub>)<sub><i>δ</i>>0</sub> is <a href="Tightness_of_measures#Exponential_tightness" title="Tightness of measures">exponentially tight</a>, then the upper bound also holds for closed sets <i>F</i>. In other words, exponential tightness enables one to convert a weak large deviation principle into a strong one.

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

<p>Naïvely, one might try to replace the two inequalities (U) and (L) by the single requirement that, for all Borel sets <i>S</i> ⊆ <i>X</i>,

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<annotation encoding="application/x-tex">{\displaystyle \lim _{\delta \downarrow 0}\delta \log \mu _{\delta }(S)=-\inf _{x\in S}I(x).\quad {\mbox{(E)}}}</annotation>

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</math></span><img src="./c1c5a3eb6d96d9659fde1a76ba0d6979b6196131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.932ex; height:4.343ex;" alt="{\displaystyle \lim _{\delta \downarrow 0}\delta \log \mu _{\delta }(S)=-\inf _{x\in S}I(x).\quad {\mbox{(E)}}}" loading="lazy"></span></dd></dl>

<p>The equality (E) is far too restrictive, since many interesting examples satisfy (U) and (L) but not (E). For example, the measure <i>μ</i><sub><i>δ</i></sub> might be <a href="Atom_(measure_theory)" title="Atom (measure theory)">non-atomic</a> for all <i>δ</i>, so the equality (E) could hold for <i>S</i> = {<i>x</i>} only if <i>I</i> were identically +∞, which is not permitted in the definition. However, the inequalities (U) and (L) do imply the equality (E) for so-called <i>I</i><b>-continuous</b> sets <i>S</i> ⊆ <i>X</i>, those for which

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<annotation encoding="application/x-tex">{\displaystyle I{\big (}{\stackrel {\circ }{S}}{\big )}=I{\big (}{\bar {S}}{\big )},}</annotation>

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</math></span><img src="./b7ce9fb547689be1c6d72c709477801b2fc1878f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.42ex; height:4.176ex;" alt="{\displaystyle I{\big (}{\stackrel {\circ }{S}}{\big )}=I{\big (}{\bar {S}}{\big )},}" 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 {\stackrel {\circ }{S}}}">

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

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</math></span><img src="./29e33afa1849dfaa74676e6c7b73f2f71c61b2bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:3.509ex;" alt="{\displaystyle {\stackrel {\circ }{S}}}" loading="lazy"></span> 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 {\bar {S}}}">

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

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</math></span><img src="./8477edc5727710564fab7b520bd09d19b1fdc95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.676ex;" alt="{\displaystyle {\bar {S}}}" loading="lazy"></span> denote the <a href="Interior_(topology)" title="Interior (topology)">interior</a> and <a href="Closure_(topology)" title="Closure (topology)">closure</a> of <i>S</i> in <i>X</i> respectively. In many examples, many sets/events of interest are <i>I</i>-continuous. For example, if <i>I</i> is a <a href="Continuous_function" title="Continuous function">continuous function</a>, then all sets <i>S</i> such that

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<annotation encoding="application/x-tex">{\displaystyle S\subseteq {\bar {\stackrel {\circ }{S}}}}</annotation>

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</math></span><img src="./e578af033eb79fee6997612ca4ab9c5143e55680.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.097ex; height:4.176ex;" alt="{\displaystyle S\subseteq {\bar {\stackrel {\circ }{S}}}}" loading="lazy"></span></dd></dl>

<p>are <i>I</i>-continuous; all open sets, for example, satisfy this containment.

</p>

<div class="mw-heading mw-heading3"><h3 id="Transformation_of_large_deviation_principles">Transformation of large deviation principles</h3></div>

<p>Given a large deviation principle on one space, it is often of interest to be able to construct a large deviation principle on another space. There are several results in this area:

</p>

<ul><li>the <a href="Contraction_principle_(large_deviations_theory)" title="Contraction principle (large deviations theory)">contraction principle</a> tells one how a large deviation principle on one space "pushes forward" (via the <a href="Pushforward_measure" title="Pushforward measure">pushforward</a> of a probability measure) to a large deviation principle on another space <i>via</i> a <a href="Continuous_function" title="Continuous function">continuous function</a>;</li>

<li>the <a href="Dawson-G%C3%A4rtner_theorem" class="mw-redirect" title="Dawson-Gärtner theorem">Dawson-Gärtner theorem</a> tells one how a sequence of large deviation principles on a sequence of spaces passes to the <a href="Projective_limit" class="mw-redirect" title="Projective limit">projective limit</a>.</li>

<li>the <a href="Tilted_large_deviation_principle" title="Tilted large deviation principle">tilted large deviation principle</a> gives a large deviation principle for integrals of exponential <a href="Functional_(mathematics)" title="Functional (mathematics)">functionals</a>.</li>

<li><a href="Exponentially_equivalent_measures" title="Exponentially equivalent measures">exponentially equivalent measures</a> have the same large deviation principles.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="History_and_basic_development">History and basic development</h2></div>

<p>The notion of a rate function emerged in the 1930s with the Swedish mathematician <a href="Harald_Cram%C3%A9r" title="Harald Cramér">Harald Cramér</a>'s study of a sequence of <b><a href="I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a></b> random variables (<i>Z</i><sub><i>i</i></sub>)<sub>i∈<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 {N} }">

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

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</math></span><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.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 {N} }" loading="lazy"></span></sub>. Namely, among some considerations of scaling, Cramér studied the behavior of the distribution of the average <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="{\textstyle X_{n}={\frac {1}{n}}\sum _{i=1}^{n}Z_{i}}">

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<annotation encoding="application/x-tex">{\textstyle X_{n}={\frac {1}{n}}\sum _{i=1}^{n}Z_{i}}</annotation>

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</math></span><img src="./a3753406aced946664b331d418598160e22a06a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.579ex; height:3.343ex;" alt="{\textstyle X_{n}={\frac {1}{n}}\sum _{i=1}^{n}Z_{i}}" loading="lazy"></span> as <i>n</i>→∞.<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> He found that the tails of the distribution of <i>X</i><sub><i>n</i></sub> decay exponentially as <i>e</i><sup>−<i>nλ</i>(<i>x</i>)</sup> where the factor <i>λ</i>(<i>x</i>) in the exponent is the Legendre–Fenchel transform (a.k.a. the <a href="Convex_conjugate" title="Convex conjugate">convex conjugate</a>) of the <a href="Cumulant" title="Cumulant">cumulant</a>-generating 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 \Psi _{Z}(t)=\log \operatorname {E} e^{tZ}.}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{Z}(t)=\log \operatorname {E} e^{tZ}.}</annotation>

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</math></span><img src="./d6a1c436eb01668cebf2fe23ad6a195aa6a648ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.049ex; height:3.176ex;" alt="{\displaystyle \Psi _{Z}(t)=\log \operatorname {E} e^{tZ}.}" loading="lazy"></span> For this reason this particular function <i>λ</i>(<i>x</i>) is sometimes called the <b>Cramér function</b>. The rate function defined above in this article is a broad generalization of this notion of Cramér's, defined more abstractly on a <a href="Probability_space" title="Probability space">probability space</a>, rather than the <a href="State_space" class="mw-redirect" title="State space">state space</a> of a random variable.

</p>

<ul><li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory</a></li>

<li>Moderate deviation principle</li></ul>

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

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</style><cite id="CITEREFCramér1938" class="citation journal cs1 cs1-prop-foreign-lang-source">Cramér, Harald (1938). "Sur un nouveau théorème-limite de la théorie des probabilités". <i>Colloque consacré à la théorie des probabilités, Part 3, Actualités scientifiques et industrielles</i> (in French). <b>731</b>: <span class="nowrap">5–</span>23.</cite></span>

</li>

</ol></div>

<ul><li><cite id="CITEREFDemboZeitouni,_Ofer1998" class="citation book cs1">Dembo, Amir; Zeitouni, Ofer (1998). <i>Large deviations techniques and applications</i>. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. xvi+396. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98406-2</bdi>.</cite> <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=1619036">1619036</a></li>

<li><cite id="CITEREFden_Hollander2000" class="citation book cs1">den Hollander, Frank (2000). <i>Large deviations</i>. <a href="Fields_Institute" title="Fields Institute">Fields Institute</a> Monographs 14. Providence, RI: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. p. x+143. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1989-5</bdi>.</cite> <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=1739680">1739680</a></li>

<li><cite id="CITEREFRassoul-AghaSeppäläinen,_Timo2015" class="citation book cs1">Rassoul-Agha, Firas; Seppäläinen, Timo (2015). <i>A course on large deviations with an introduction to Gibbs measures</i>. Graduate Studies in Mathematics 162. Providence, RI: American Mathematical Society. xiv+318. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-7578-0</bdi>.</cite> <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=3309619">3309619</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-01-25" href="https://en.wikipedia.org/wiki/?title=Rate_function&oldid=1199026965">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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