{"id":1031,"date":"2024-01-01T13:48:06","date_gmt":"2024-01-01T23:48:06","guid":{"rendered":"https:\/\/blogs.oregonstate.edu\/tpham\/?p=1031"},"modified":"2024-03-19T00:55:15","modified_gmt":"2024-03-19T10:55:15","slug":"characterization-degenerate-scaling","status":"publish","type":"post","link":"https:\/\/blogs.oregonstate.edu\/tpham\/characterization-degenerate-scaling\/","title":{"rendered":"Characterization of degenerate scaling"},"content":{"rendered":"\n

This post is a follow-up to my post on 12\/29\/2023<\/a>, where I posed the following question:<\/p>\n\n\n\n

\n

Let (X_n) be an i.i.d. sequence of \\mathbb{R}-valued random variables. Let (a_n) be a positive sequence such that \\lim a_n=\\infty. Under what condition of (a_n) can one conclude that \\lim\\frac{X_n}{a_n}=0 almost surely?<\/p>\n<\/blockquote>\n\n\n\n

I have found a full characterization of the sequence (a_n). After all, it is an interesting exercise.<\/p>\n\n\n\n

\n

Proposition:<\/strong> Let (X_n) be an i.i.d. sequence of \\mathbb{R}-valued random variables Let (a_n) be a positive sequence such that \\lim a_n=\\infty. If \\sum P\\left(|X_1|>\\frac{a_n}{c}\\right)<\\infty for any constant c>0, then \\lim\\frac{X_n}{a_n}=0 almost surely. Otherwise, almost surely \\frac{X_n}{a_n} does not converge.<\/em><\/p>\n<\/blockquote>\n\n\n\n

As an application, if (X_n) is an i.i.d. sequence of mean-one exponentially distributed random variables, then almost surely the sequence of \\frac{X_n}{\\ln n} does not converge.<\/p>\n\n\n\n

Proof:<\/strong> First, suppose that there exists c>0 such that \\sum P\\left(|X_1|>\\frac{a_n}{c}\\right)=\\infty. Then the independent events E_n=[|X_n|>\\frac{a_n}{c}] satisfies<\/p>\n\n\n\n

\\displaystyle \\sum \\mathbb{P}(E_n)=\\infty.\n\n\n\n

By Borel-Cantelli’s Lemma, almost surely the events E_n occurs for infinitely many n. Therefore, with probability 1, the sequence (X_n\/a_n) does not converge to 0. Now fix b\\neq 0. We show that almost surely X_n\/a_n\\not\\to b. Consider the independent events E'_n=\\left[|X_n|<\\frac{|b|a_n}{2}\\right]. One has<\/p>\n\n\n\n

\\displaystyle \\sum\\mathbb{P}(E'_n)=\\infty<\/p>\n\n\n\n

By Borel-Cantelli’s Lemma, almost surely the events E'_n occurs for infinitely many n. Therefore, with probability 1, the sequence (X_n\/a_n) does not converge to b.<\/p>\n\n\n\n

Next, suppose that for every c>0, one has \\sum P\\left(|X_1|>\\frac{a_n}{c}\\right)<\\infty. For a fixed constant c>0, the event A_c=\\left[\\limsup\\frac{|X_n|}{a_n}\\le \\frac{1}{c}\\right] is a tail event. Thus, \\mathbb{P}(A_c)=0 or \\mathbb{P}(A_c)=1 according to Kolmogorov’s 0-1 Law. Note that \\cap_{n=1}^\\infty \\left[|X_n|\\le \\frac{a_n}{c}\\right]\\subset A_c and <\/p>\n\n\n\n

\\displaystyle \\mathbb{P}\\left(\\bigcap\\limits_{n=1}^\\infty \\left[|X_n|\\le \\frac{a_n}{c}\\right]\\right)=\\prod_{n=1}^\\infty \\left(1-P\\left(|X_1|>\\frac{a_n}{c}\\right)\\right)<\/p>\n\n\n\n

which is positive because \\sum P\\left(|X_1|>\\frac{a_n}{c}\\right)<\\infty. Thus, \\mathbb{P}(A_c)>0 and therefore, \\mathbb{P}(A_c)=1. Now note that<\/p>\n\n\n\n

\\displaystyle \\mathbb{P}\\left(\\lim\\frac{X_n}{a_n}=0\\right)= \\mathbb{P}\\left(\\limsup\\frac{|X_n|}{a_n}=0\\right)=\\lim_{k\\to\\infty} \\mathbb{P}(A_{k})=1.<\/p>\n\n\n\n

This completes the proof.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post is a follow-up to my post on 12\/29\/2023. I have found a full characterization of the sequence $latex (a_n).$ After all, it is an interesting exercise. Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":12738,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[14],"class_list":["post-1031","post","type-post","status-publish","format-standard","hentry","category-math","tag-probability"],"_links":{"self":[{"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts\/1031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/users\/12738"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/comments?post=1031"}],"version-history":[{"count":38,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts\/1031\/revisions"}],"predecessor-version":[{"id":1182,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts\/1031\/revisions\/1182"}],"wp:attachment":[{"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/media?parent=1031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/categories?post=1031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/tags?post=1031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}