{"id":183165,"date":"2007-01-08T00:00:00","date_gmt":"2009-10-31T10:24:05","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/upcrossing-inequalities-for-stationary-sequences\/"},"modified":"2016-09-09T09:48:29","modified_gmt":"2016-09-09T16:48:29","slug":"upcrossing-inequalities-for-stationary-sequences","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/upcrossing-inequalities-for-stationary-sequences\/","title":{"rendered":"Upcrossing inequalities for stationary sequences"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Let <i>g<\/i> be a function which assigns to each stationary process <i>X<sub>n<\/sub><\/i> and to each sample <i>X<sub>1<\/sub>, X<sub>2<\/sub> ,&#8230;, X<sub>n<\/sub><\/i> of the process a real number <i>g(X<sub>1<\/sub>, &#8230; ,X<sub>n<\/sub>)<\/i>, which may also depend on the distribution of the process. I&#8217;ll discuss how one can obtain effective bounds on the probability that the sequence <i>g(X<sub>1<\/sub>, &#8230;, X<sub>n<\/sub>)<\/i> crosses a fixed interval some number of times in terms of a quantity measuring the &#8220;average sub-additivity&#8221; of <i>g<\/i>.  Applications include universal upcrossing inequalities for Kingman&#8217;s sub-additive ergodic theorem, for the Shannon-McMillan-Breiman theorem, and for the Kolmogorov complexity statistic.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let g be a function which assigns to each stationary process Xn and to each sample X1, X2 ,&#8230;, Xn of the process a real number g(X1, &#8230; ,Xn), which may also depend on the distribution of the process. I&#8217;ll discuss how one can obtain effective bounds on the probability that the sequence g(X1, &#8230;, [&hellip;]<\/p>\n","protected":false},"featured_media":194929,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-183165","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/FrSyF04oehs","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183165","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/183165\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194929"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=183165"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=183165"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=183165"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=183165"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=183165"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=183165"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=183165"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=183165"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=183165"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=183165"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}