{"id":6797,"date":"2014-02-12T22:10:13","date_gmt":"2014-02-12T21:10:13","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=6797"},"modified":"2014-09-13T09:42:23","modified_gmt":"2014-09-13T07:42:23","slug":"back-to-basics-a-moment-with-moments","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2014\/02\/12\/back-to-basics-a-moment-with-moments\/","title":{"rendered":"Back to basics - A moment with moments"},"content":{"rendered":"

 <\/p>\n

\"Teaching\"<\/a><\/p>\n

 <\/p>\n

I like very much the following elementary observation, inspired from a lemma taken from an article by Kabluchko and Zaporozhets<\/a>. Let $X_1,X_2,\\ldots$ be independent and identically distributed random variables, and let $\\Phi:\\mathbb{R}_+\\to\\mathbb{R}_+$ be some increasing function with (increasing) inverse $\\Phi^{-1}:\\mathbb{R}_+\\to\\mathbb{R}_+$. Examples: $\\Phi(t)=t^p$,\u00a0 or $\\Phi(t)=\\log(1+t)$. Then<\/p>\n

$$\\mathbb{E}(\\Phi(|X_1|))<\\infty \\quad\\Leftrightarrow\\quad \\mathbb{P}\\left(\\frac{|X_n|}{\\Phi^{-1}(n)}<1\\mbox{ for } n\\gg1\\right)=1.$$<\/p>\n

To see it, recall that $\\sum_{n\\geq1}\\mathbf{1}_{n\\leq y} =\\lfloor y\\rfloor\\leq y\\leq1+\\lfloor y\\rfloor=\\sum_{n=0}^\\infty\\mathbf{1}_{n\\leq y}$ for every non-negative real number $y\\geq0$, and thus, if $Y\\geq0$ is a non-negative random variable then<\/p>\n

$$\\sum_{n=1}^\\infty\\mathbb{P}(Y\\geq n)\\leq\\mathbb{E}(Y)\\leq\\sum_{n=0}^\\infty\\mathbb{P}(Y\\geq n).$$<\/p>\n

Used with $Y=\\Phi(|X_1|)$, we obtain,<\/p>\n

$$\\mathbb{E}(\\Phi(|X_1|)<\\infty\\quad\\Leftrightarrow\\quad\\sum_{n=1}^\\infty\\mathbb{P}(|X_1|\\geq \\Phi^{-1}(n))<\\infty.$$<\/p>\n

Now $X_1\\overset{d}{=}X_n$, and since $X_1,X_2,\\ldots $ are independent, the Borel-Cantelli lemmas give<\/p>\n

$$\\sum_{n\\geq1}\\mathbb{P}(|X_n|\\geq\\Phi^{-1}(n))<\\infty\\quad\\Leftrightarrow\\quad\\mathbb{P}\\left(\\frac{|X_n|}{\\Phi^{-1}(n)}<1\\mbox{ for }n\\gg1\\right)=1.$$<\/p>\n","protected":false},"excerpt":{"rendered":"

    I like very much the following elementary observation, inspired from a lemma taken from an article by Kabluchko and Zaporozhets. Let $X_1,X_2,\\ldots$ be…<\/p>\n

Continue readingBack to basics - A moment with moments<\/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":103},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/6797"}],"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=6797"}],"version-history":[{"count":49,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/6797\/revisions"}],"predecessor-version":[{"id":7762,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/6797\/revisions\/7762"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=6797"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=6797"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=6797"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}