{"title":"在另一个LIL上,对于没有有限方差的变量","authors":"R. Pakshirajan, M. Sreehari","doi":"10.1090/tpms/1179","DOIUrl":null,"url":null,"abstract":"<p>In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X Subscript n\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mi>n</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">X_n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> satisfies the condition that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"limit Underscript x right-arrow normal infinity Endscripts StartFraction log upper H left-parenthesis x right-parenthesis Over left-parenthesis log x right-parenthesis Superscript delta Baseline EndFraction equals 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:munder>\n <mml:mo movablelimits=\"true\" form=\"prefix\">lim</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi>\n </mml:mrow>\n </mml:munder>\n <mml:mfrac>\n <mml:mrow>\n <mml:mi>log</mml:mi>\n <mml:mo><!-- --></mml:mo>\n <mml:mi>H</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>log</mml:mi>\n <mml:mo><!-- --></mml:mo>\n <mml:mi>x</mml:mi>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mi>δ<!-- δ --></mml:mi>\n </mml:msup>\n </mml:mrow>\n </mml:mfrac>\n <mml:mo>=</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\lim _{ x\\rightarrow \\infty } \\frac {\\log H(x)}{(\\log x)^\\delta } = 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for some <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than delta greater-than 1 slash 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>></mml:mo>\n <mml:mi>δ<!-- δ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">0 >\\delta > 1/2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H left-parenthesis x right-parenthesis equals sans-serif upper E left-parenthesis upper X 1 squared upper I left-parenthesis StartAbsoluteValue upper X 1 EndAbsoluteValue less-than-or-equal-to x right-parenthesis right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>H</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">E</mml:mi>\n </mml:mrow>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:msubsup>\n <mml:mi>X</mml:mi>\n <mml:mn>1</mml:mn>\n <mml:mn>2</mml:mn>\n </mml:msubsup>\n <mml:mi>I</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:msub>\n <mml:mi>X</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">|</mml:mo>\n </mml:mrow>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H(x)=\\mathsf E\\left (X_1^2 I(|X_1|\\le x)\\right )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a slowly varying function. The condition above is not very restrictive.</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the other LIL for variables without finite variance\",\"authors\":\"R. Pakshirajan, M. Sreehari\",\"doi\":\"10.1090/tpms/1179\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X Subscript n\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>X</mml:mi>\\n <mml:mi>n</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">X_n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> satisfies the condition that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"limit Underscript x right-arrow normal infinity Endscripts StartFraction log upper H left-parenthesis x right-parenthesis Over left-parenthesis log x right-parenthesis Superscript delta Baseline EndFraction equals 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:munder>\\n <mml:mo movablelimits=\\\"true\\\" form=\\\"prefix\\\">lim</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">→<!-- → --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">∞<!-- ∞ --></mml:mi>\\n </mml:mrow>\\n </mml:munder>\\n <mml:mfrac>\\n <mml:mrow>\\n <mml:mi>log</mml:mi>\\n <mml:mo><!-- --></mml:mo>\\n <mml:mi>H</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>log</mml:mi>\\n <mml:mo><!-- --></mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mi>δ<!-- δ --></mml:mi>\\n </mml:msup>\\n </mml:mrow>\\n </mml:mfrac>\\n <mml:mo>=</mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lim _{ x\\\\rightarrow \\\\infty } \\\\frac {\\\\log H(x)}{(\\\\log x)^\\\\delta } = 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> for some <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"0 greater-than delta greater-than 1 slash 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>0</mml:mn>\\n <mml:mo>></mml:mo>\\n <mml:mi>δ<!-- δ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">0 >\\\\delta > 1/2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper H left-parenthesis x right-parenthesis equals sans-serif upper E left-parenthesis upper X 1 squared upper I left-parenthesis StartAbsoluteValue upper X 1 EndAbsoluteValue less-than-or-equal-to x right-parenthesis right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>H</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"sans-serif\\\">E</mml:mi>\\n </mml:mrow>\\n <mml:mrow>\\n <mml:mo>(</mml:mo>\\n <mml:msubsup>\\n <mml:mi>X</mml:mi>\\n <mml:mn>1</mml:mn>\\n <mml:mn>2</mml:mn>\\n </mml:msubsup>\\n <mml:mi>I</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:msub>\\n <mml:mi>X</mml:mi>\\n <mml:mn>1</mml:mn>\\n </mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">|</mml:mo>\\n </mml:mrow>\\n <mml:mo>≤<!-- ≤ --></mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>)</mml:mo>\\n </mml:mrow>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">H(x)=\\\\mathsf E\\\\left (X_1^2 I(|X_1|\\\\le x)\\\\right )</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is a slowly varying function. The condition above is not very restrictive.</p>\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1179\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1179","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
On the other LIL for variables without finite variance
In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables XnX_n satisfies the condition that limx→∞logH(x)(logx)δ=0\lim _{ x\rightarrow \infty } \frac {\log H(x)}{(\log x)^\delta } = 0 for some 0>δ>1/20 >\delta > 1/2, where H(x)=E(X12I(|X1|≤x))H(x)=\mathsf E\left (X_1^2 I(|X_1|\le x)\right ) is a slowly varying function. The condition above is not very restrictive.