A computable bound of the essential spectral radius of finite range Metropolis-Hastings kernels

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• 作者：Loï ; c Hervé ; ^{Loic.Herve@insa-rennes.fr" class="auth_mail" title="E-mail the corresponding author} ; James Ledoux ; ^{James.Ledoux@insa-rennes.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60J10 ; 47B07
• 刊名：Statistics & Probability Letters
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：117
• 期：Complete
• 页码：72-79
• 全文大小：417 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si1.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=aa82a7dded6f8c9ec89e2a3dba7c8cbb" title="Click to view the MathML source">πclass="mathContainer hidden">class="mathCode">$\pi$ be a positive continuous target density on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si2.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=5909e71ebace475fd405fd2be08510f9" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$. Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si3.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=26bcdced0c26f13df9dcede512eb6345" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$P$ be the Metropolis–Hastings operator on the Lebesgue space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si4.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=d3c11f1dbd054d00b4a0ffe238b0ef9a" title="Click to view the MathML source">L²(π)class="mathContainer hidden">class="mathCode">${\mathbb{L}}^2\left(\pi \right)$ corresponding to a proposal Markov kernel class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si5.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=407bc19e3e1bc6c27ea24c7aa986b8ac" title="Click to view the MathML source">Qclass="mathContainer hidden">class="mathCode">$Q$ on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si2.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=5909e71ebace475fd405fd2be08510f9" title="Click to view the MathML source">Rclass="mathContainer hidden">class="mathCode">$\mathbb{R}$. When using the quasi-compactness method to estimate the spectral gap of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si3.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=26bcdced0c26f13df9dcede512eb6345" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$P$, a mandatory first step is to obtain an accurate bound of the essential spectral radius class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si8.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=4653b84015949fd6e102634393fd2bc4" title="Click to view the MathML source">r_ess(P)class="mathContainer hidden">class="mathCode">$r_{ess}\left(P\right)$ of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si3.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=26bcdced0c26f13df9dcede512eb6345" title="Click to view the MathML source">Pclass="mathContainer hidden">class="mathCode">$P$. In this paper a computable bound of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si8.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=4653b84015949fd6e102634393fd2bc4" title="Click to view the MathML source">r_ess(P)class="mathContainer hidden">class="mathCode">$r_{ess}\left(P\right)$ is obtained under the following assumption on the proposal kernel: class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si5.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=407bc19e3e1bc6c27ea24c7aa986b8ac" title="Click to view the MathML source">Qclass="mathContainer hidden">class="mathCode">$Q$ has a bounded continuous density class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si12.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=1ad746e4510b4ed9663561b4983955ac" title="Click to view the MathML source">q(x,y)class="mathContainer hidden">class="mathCode">$q(x,y)$ on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si13.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=af78ac92b65d96e423994cb4eec73d03" title="Click to view the MathML source">R²class="mathContainer hidden">class="mathCode">${\mathbb{R}}^2$ satisfying the following finite range assumption : class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si14.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=433ef4d8cd28d1518ebab0d066dad755">class="imgLazyJSB inlineImage" height="15" width="180" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216300608-si14.gif">class="mathContainer hidden">class="mathCode">$\left|u\right|>s\Rightarrow q(x,x+u)=0$ (for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167715216300608&_mathId=si15.gif&_user=111111111&_pii=S0167715216300608&_rdoc=1&_issn=01677152&md5=e702ccb5bb0e1920be9e2692589fa11b" title="Click to view the MathML source">s>0class="mathContainer hidden">class="mathCode">$s>0$). This result is illustrated with Random Walk Metropolis–Hastings kernels.