{"id":22,"date":"2010-04-30T17:06:32","date_gmt":"2010-04-30T15:06:32","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=22"},"modified":"2019-11-10T19:04:10","modified_gmt":"2019-11-10T18:04:10","slug":"convergence-of-random-matrices-and-moments","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2010\/04\/30\/convergence-of-random-matrices-and-moments\/","title":{"rendered":"Star moments convergence of random matrices"},"content":{"rendered":"<p style=\"text-align: justify;\">Let $(X_{jk})_{j,k\\geq1}$ be an infinite table of i.i.d. complex random variables with positive variance and all moments bounded by a constant. Set $M:=(X_{jk})_{1\\leq j,k\\leq n}$. I do believe that with probability one and in expectation, the $*$-moments of $M$ converge as $n\\to\\infty$ to the $*$-moments of the Voiculescu circular element. This result is well known when $X_{11}$ is Gaussian. There is maybe a proof written somewhere, involving some paths combinatorics. Can you link the question with the work of <a title=\"Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs\" href=\"http:\/\/arxiv.org\/abs\/1002.1212\">Nourdin and Peccati<\/a>?<\/p>\n<p style=\"text-align: justify;\">If it holds, this statement shows that the hypothesis in the <a href=\"http:\/\/www.ams.org\/mathscinet\/search\/publications.html?pg1=IID&amp;s1=363478\">\u015a<\/a><a title=\"Random regularization of the Brown spectral measure \" href=\"http:\/\/www.ams.org\/mathscinet-getitem?mr=1929504\">niady theorem<\/a> is always satisfied by these random matrices. Moreover, the regularization by a Ginibre Ensemble is not needed thanks to the Tao and Vu bound on the smallest singular values. Well, TT will probably say a word on this in <a title=\"Terence Tao course on random matrices\" href=\"http:\/\/en.wordpress.com\/tag\/254a-random-matrices\/\">his recent course<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let $(X_{jk})_{j,k\\geq1}$ be an infinite table of i.i.d. complex random variables with positive variance and all moments bounded by a constant. Set $M:=(X_{jk})_{1\\leq j,k\\leq n}$.&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2010\/04\/30\/convergence-of-random-matrices-and-moments\/\">Continue reading<span class=\"screen-reader-text\">Star moments convergence of random matrices<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":66},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=22"}],"version-history":[{"count":1,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22\/revisions"}],"predecessor-version":[{"id":11753,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/22\/revisions\/11753"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=22"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=22"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=22"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}