{"id":974,"date":"2023-12-29T14:47:13","date_gmt":"2023-12-30T00:47:13","guid":{"rendered":"https:\/\/blogs.oregonstate.edu\/tpham\/?p=974"},"modified":"2024-03-19T00:58:55","modified_gmt":"2024-03-19T10:58:55","slug":"degenerate-scaling","status":"publish","type":"post","link":"https:\/\/blogs.oregonstate.edu\/tpham\/degenerate-scaling\/","title":{"rendered":"Degenerate scaling of random variables"},"content":{"rendered":"\n

There are well-known results about the scaling and centering laws of a sequence of \\mathbb{R}-valued random variables. That is the problem of finding the right coefficients a_n and b_n such that the sequence \\frac{X_n-b_n}{a_n} converges in law. See, for example, Section 14.8 of [1]<\/a>. Normally, one tries to avoid the degenerate type, which is when the limit distribution is a Dirac mass at 0. I recently ran into a problem where the degenerate type is desirable. More specifically:<\/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

This problem seems to be quite classic. If X_1 has a bounded range, i.e. \\mathbb{P}(X_1\\in[a,b])=1 for some numbers a and b, then no additional condition of (a_n) is needed. How about the case X_1 has an unbounded range? This is essentially Problem 51 on page 263 of [1]<\/a>. The textbook does not ask for a full characterization of the sequence a_n. I find the problem quite interesting.<\/p>\n\n\n\n

The first observation is that the event A=\\left[\\lim\\frac{X_n}{a_n}=0\\right] is a tail event. By Kolmogorov’s 0-1 Law<\/a>, either \\mathbb{P}(A)=0 or \\mathbb{P}(A)=1. We just need to decide between the two. One may be tempted to think that \\frac{X_n}{a_n} is essentially the same as \\frac{X_1}{a_n}, so it should almost surely converge 0 as n\\to\\infty. This argument is not correct because \\frac{X_n}{a_n} only has the same distribution as \\frac{X_1}{a_n}. If the range of values of X_1 is unbounded, then so is the range of values of \\frac{X_1}{a_n}. Let us consider two following cases, one of which gives \\mathbb{P}(A)=0 and the other gives \\mathbb{P}(A)=1. Let us denote the cumulative distribution function of |X_1| by F(x).<\/p>\n\n\n\n

Case 1:<\/strong> there exists a positive sequence (c_n) such that<\/p>\n\n\n\n

\\displaystyle \\lim\\frac{c_n}{a_n}=0 and \\displaystyle\\sum(1-F(c_n))<\\infty.<\/p>\n\n\n\n

Let E_n=[|X_n|\\le c_n] and E=\\cap_{n=1}^\\infty E_n. Then \\mathbb{P}(E_n)=F(c_n) and<\/p>\n\n\n\n

\\displaystyle \\mathbb{P}(E)=\\prod_{n=1}^\\infty F(c_n)=\\prod_{n=1}^\\infty (1-(1-F(c_n))<\/p>\n\n\n\n

which has a finite positive value because \\sum(1-F(c_n))<\\infty. It is easy to see that E\\subset A and thus \\mathbb{P}(A)=1.<\/p>\n\n\n\n

Case 2:<\/strong> there exists a positive sequence (c_n) such that<\/p>\n\n\n\n

\\displaystyle \\lim\\frac{c_n}{a_n}>0 and \\displaystyle\\sum(1-F(c_n))=\\infty.<\/p>\n\n\n\n

Let E_n=[|X_n|> c_n] and E=\\limsup E_n. The events E_n are independent and<\/p>\n\n\n\n

\\displaystyle \\sum \\mathbb{P}(E_n)=\\sum(1-F(c_n))=\\infty.\n\n\n\n

By Borel-Cantelli’s Lemma<\/a>, \\mathbb{P}(E)=1. It is easy to see that E\\subset A^c. Thus, \\mathbb{P}(A)=1-\\mathbb{P}(A^c)=0.<\/p>\n\n\n\n

Final thoughts:<\/strong> Cases 1 and 2 do not cover all possibilities. One can see this through the simple case F(x)=1-e^{-x} (the case when the random variables X_n are i.i.d. exponentials of mean one) and a_n=2\\ln n. It would be interesting to see a full characterization of the sequence (a_n) for which one has \\mathbb{P}(A)=1. I think this should be known somewhere in the literature.<\/p>\n\n\n\n

Update 01\/01\/2024<\/a>: I have resolved the problem.<\/p>\n\n\n\n

References:<\/h2>\n
    \n
  1. “A modern approach to Probability Theory<\/em>” (1997) by Friestedt and Gray.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Let $latex (X_n)$ be an i.i.d. sequence of $latex \\mathbb{R}$-valued random variables. Let $latex (a_n)$ be a positive sequence such that $latex \\lim a_n=\\infty$. Under what condition of $latex (a_n)$ can one conclude that $latex \\lim\\frac{X_n}{a_n}=0$ almost surely? 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-974","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\/974","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=974"}],"version-history":[{"count":54,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts\/974\/revisions"}],"predecessor-version":[{"id":1183,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/posts\/974\/revisions\/1183"}],"wp:attachment":[{"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/media?parent=974"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/categories?post=974"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.oregonstate.edu\/tpham\/wp-json\/wp\/v2\/tags?post=974"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}