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

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• 作者：Lo&iuml ; 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 lsi1" class="mathmlsrc">lass="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">πlass="mathContainer hidden">lass="mathCode">low="scroll">π be a positive continuous target density on lsi2" class="mathmlsrc">lass="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">Rlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">R. Let lsi3" class="mathmlsrc">lass="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">Plass="mathContainer hidden">lass="mathCode">low="scroll">P be the Metropolis–Hastings operator on the Lebesgue space lsi4" class="mathmlsrc">lass="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)lass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">L2(π) corresponding to a proposal Markov kernel lsi5" class="mathmlsrc">lass="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">Qlass="mathContainer hidden">lass="mathCode">low="scroll">Q on lsi2" class="mathmlsrc">lass="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">Rlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">R. When using the quasi-compactness method to estimate the spectral gap of lsi3" class="mathmlsrc">lass="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">Plass="mathContainer hidden">lass="mathCode">low="scroll">P, a mandatory first step is to obtain an accurate bound of the essential spectral radius lsi8" class="mathmlsrc">lass="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)lass="mathContainer hidden">lass="mathCode">low="scroll">ress(P) of lsi3" class="mathmlsrc">lass="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">Plass="mathContainer hidden">lass="mathCode">low="scroll">P. In this paper a computable bound of lsi8" class="mathmlsrc">lass="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)lass="mathContainer hidden">lass="mathCode">low="scroll">ress(P) is obtained under the following assumption on the proposal kernel: lsi5" class="mathmlsrc">lass="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">Qlass="mathContainer hidden">lass="mathCode">low="scroll">Q has a bounded continuous density lsi12" class="mathmlsrc">lass="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)lass="mathContainer hidden">lass="mathCode">low="scroll">q(x,y) on lsi13" class="mathmlsrc">lass="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">R²lass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">R2 satisfying the following finite range assumption : lsi14" class="mathmlsrc">le="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">lass="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">lass="mathContainer hidden">lass="mathCode">low="scroll">|u|>s⇒q(x,x+u)=0 (for some lsi15" class="mathmlsrc">lass="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>0lass="mathContainer hidden">lass="mathCode">low="scroll">s>0). This result is illustrated with Random Walk Metropolis–Hastings kernels.