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<p>stub sort</p> <p><b>New page</b></p><div>In [[mathematics]], the notion of &#039;&#039;&#039;exponential equivalence of measures&#039;&#039;&#039; is a concept that describes how two sequences or families of [[probability measure]]s are “the same” from the point of view of [[large deviations theory]].<br /> <br /> ==Definition==<br /> <br /> Let (&#039;&#039;M&#039;&#039;,&amp;nbsp;&#039;&#039;d&#039;&#039;) be a [[metric space]] and consider two one-[[parameter]] families of probability measures on &#039;&#039;M&#039;&#039;, say (&#039;&#039;&amp;mu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt; and (&#039;&#039;&amp;nu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt;. These two families are said to be &#039;&#039;&#039;exponentially equivalent&#039;&#039;&#039; if there exist<br /> * a one-parameter family of probability spaces ((Ω,&amp;nbsp;Σ&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;,&amp;nbsp;&#039;&#039;&#039;P&#039;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;))&lt;sub&gt;&#039;&#039;&amp;epsilon;&amp;gt;0&lt;/sub&gt;,<br /> * two families of &#039;&#039;M&#039;&#039;-valued random variables (&#039;&#039;Y&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;ε&#039;&#039;&amp;gt;0&lt;/sub&gt; and (&#039;&#039;Z&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;ε&#039;&#039;&amp;gt;0&lt;/sub&gt;,<br /> such that<br /> * for each &#039;&#039;ε&#039;&#039;&amp;nbsp;&amp;gt;&amp;nbsp;0, the &#039;&#039;&#039;P&#039;&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;-law (i.e. the [[push-forward measure]]) of &#039;&#039;Y&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt; is &#039;&#039;μ&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;, and the &#039;&#039;&#039;P&#039;&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;-law of &#039;&#039;Z&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt; is &#039;&#039;ν&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;,<br /> * for each &#039;&#039;δ&#039;&#039;&amp;nbsp;&amp;gt;&amp;nbsp;0, &amp;ldquo;&#039;&#039;Y&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt; and &#039;&#039;Z&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt; are further than &#039;&#039;δ&#039;&#039; apart&amp;rdquo; is a &amp;Sigma;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;-[[measurable set|measurable event]], i.e.<br /> ::&lt;math&gt;\big\{ \omega \in \Omega \big| d(Y_{\varepsilon}(\omega), Z_{\varepsilon}(\omega)) &gt; \delta \big\} \in \Sigma_{\varepsilon},&lt;/math&gt;<br /> * for each &#039;&#039;δ&#039;&#039;&amp;nbsp;&amp;gt;&amp;nbsp;0,<br /> ::&lt;math&gt;\limsup_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P}_{\varepsilon} \big[ d(Y_{\varepsilon}, Z_{\varepsilon}) &gt; \delta \big] = - \infty.&lt;/math&gt;<br /> <br /> The two families of random variables (&#039;&#039;Y&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;ε&#039;&#039;&amp;gt;0&lt;/sub&gt; and (&#039;&#039;Z&#039;&#039;&lt;sub&gt;&#039;&#039;ε&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;ε&#039;&#039;&amp;gt;0&lt;/sub&gt; are also said to be &#039;&#039;&#039;exponentially equivalent&#039;&#039;&#039;.<br /> <br /> ==Properties==<br /> <br /> The main use of exponential equivalence is that as far as large deviations principles are concerned, exponentially equivalent families of measures are indistinguishable. More precisely, if a large deviations principle holds for (&#039;&#039;&amp;mu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt; with good [[rate function]] &#039;&#039;I&#039;&#039;, and (&#039;&#039;&amp;mu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt; and (&#039;&#039;&amp;nu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt; are exponentially equivalent, then the same large deviations principle holds for (&#039;&#039;&amp;nu;&#039;&#039;&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&lt;/sub&gt;)&lt;sub&gt;&#039;&#039;&amp;epsilon;&#039;&#039;&amp;gt;0&lt;/sub&gt; with the same good rate function &#039;&#039;I&#039;&#039;.<br /> <br /> ==References==<br /> <br /> * {{cite book<br /> | last= Dembo<br /> | first = Amir<br /> | coauthors = Zeitouni, Ofer<br /> | title = Large deviations techniques and applications<br /> | series = Applications of Mathematics (New York) 38<br /> | edition = Second edition<br /> | publisher = Springer-Verlag<br /> | location = New York<br /> | year = 1998<br /> | pages = xvi+396<br /> | isbn = 0-387-98406-2<br /> | mr = 1619036<br /> }} (See section 4.2.2)<br /> <br /> [[Category:Asymptotic analysis]]<br /> [[Category:Probability theory]]</div>

en>David Eppstein