A supplement to the Einmahl-Li results on two-sided iterated logarithm type behavior for i.i.d. random variables

Deli Li; Yongcheng Qi; Andrew Rosalsky

doi:10.1080/07362990802286566

A supplement to the Einmahl-Li results on two-sided iterated logarithm type behavior for i.i.d. random variables

Deli Li, Yongcheng Qi, Andrew Rosalsky

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

For a sequence of i.i.d. mean 0 random variables {X, X_n; n ≥ 1} with partial sums equation presented, necessary and/or sufficient conditions are provided for {X, X_n; n ≥ 1} to enjoy iterated logarithm type behavior of the form equation presented almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases IEX² < ∞ and IEX2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an "integral test" involving whether equation presented is finite or infinite where Lx = log_e(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).

Original language	English (US)
Pages (from-to)	1095-1110
Number of pages	16
Journal	Stochastic Analysis and Applications
Volume	26
Issue number	5
DOIs	https://doi.org/10.1080/07362990802286566
State	Published - Sep 2008

Bibliographical note

Funding Information:
1Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada 2Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota, USA 3Department of Statistics, University of Florida, Gainesville, Florida, USA Abstract: For a sequence∑ of i.i.d. mean 0 random variables X Xn n ≥ 1 with partial sums Sn = i=1n Xi n ≥ 1, necessary and/or sufficient conditions are provided for X Xn n ≥ 1√ to enjoy iterated logarithm type behavior of the form 0 < lim supn→ Sn / nh n < almost surely where h · is a positive, nondecreasing function that is slowly varying at infinity. New results are obtained for the cases X2< and X2= . The proofs rely heavily on Received March 18, 2008; Accepted March 21, 2008 The authors are extremely grateful to Professor Uwe Einmahl for his interest in our work and for his careful reading of the manuscript. Professor Einmahl so kindly offered numerous substantial suggestions for improving the article. Our original version of Lemma 1 had pertained only to the particular functions h x = LLx p p ≥ 1 and h x = Lx r r > 0. Professor Einmahl suggested that we go and obtain a version of Lemma 1 for more general h · in the form of an “integral test.” Our new version of Lemma 1 has thus enabled us to present the very general Theorems 3 and 4 rather than the special cases (as in Corollaries 1 and 2) which we originally had presented. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176.

Keywords

Almost surely
Cluster set
Iterated logarithm type behavior
Law of the iterated logarithm
Slowly varying function
Sums of i.i.d. random variables

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title = "A supplement to the Einmahl-Li results on two-sided iterated logarithm type behavior for i.i.d. random variables",

abstract = "For a sequence of i.i.d. mean 0 random variables {X, Xn; n ≥ 1} with partial sums equation presented, necessary and/or sufficient conditions are provided for {X, Xn; n ≥ 1} to enjoy iterated logarithm type behavior of the form equation presented almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases IEX2 < ∞ and IEX2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an {"}integral test{"} involving whether equation presented is finite or infinite where Lx = loge(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).",

keywords = "Almost surely, Cluster set, Iterated logarithm type behavior, Law of the iterated logarithm, Slowly varying function, Sums of i.i.d. random variables",

author = "Deli Li and Yongcheng Qi and Andrew Rosalsky",

note = "Funding Information: 1Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada 2Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota, USA 3Department of Statistics, University of Florida, Gainesville, Florida, USA Abstract: For a sequence∑ of i.i.d. mean 0 random variables X Xn n ≥ 1 with partial sums Sn = i=1n Xi n ≥ 1, necessary and/or sufficient conditions are provided for X Xn n ≥ 1√ to enjoy iterated logarithm type behavior of the form 0 < lim supn→ Sn / nh n < almost surely where h · is a positive, nondecreasing function that is slowly varying at infinity. New results are obtained for the cases X2< and X2= . The proofs rely heavily on Received March 18, 2008; Accepted March 21, 2008 The authors are extremely grateful to Professor Uwe Einmahl for his interest in our work and for his careful reading of the manuscript. Professor Einmahl so kindly offered numerous substantial suggestions for improving the article. Our original version of Lemma 1 had pertained only to the particular functions h x = LLx p p ≥ 1 and h x = Lx r r > 0. Professor Einmahl suggested that we go and obtain a version of Lemma 1 for more general h · in the form of an “integral test.” Our new version of Lemma 1 has thus enabled us to present the very general Theorems 3 and 4 rather than the special cases (as in Corollaries 1 and 2) which we originally had presented. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176.",

year = "2008",

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language = "English (US)",

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journal = "Stochastic Analysis and Applications",

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T1 - A supplement to the Einmahl-Li results on two-sided iterated logarithm type behavior for i.i.d. random variables

AU - Li, Deli

AU - Qi, Yongcheng

AU - Rosalsky, Andrew

N1 - Funding Information: 1Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada 2Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota, USA 3Department of Statistics, University of Florida, Gainesville, Florida, USA Abstract: For a sequence∑ of i.i.d. mean 0 random variables X Xn n ≥ 1 with partial sums Sn = i=1n Xi n ≥ 1, necessary and/or sufficient conditions are provided for X Xn n ≥ 1√ to enjoy iterated logarithm type behavior of the form 0 < lim supn→ Sn / nh n < almost surely where h · is a positive, nondecreasing function that is slowly varying at infinity. New results are obtained for the cases X2< and X2= . The proofs rely heavily on Received March 18, 2008; Accepted March 21, 2008 The authors are extremely grateful to Professor Uwe Einmahl for his interest in our work and for his careful reading of the manuscript. Professor Einmahl so kindly offered numerous substantial suggestions for improving the article. Our original version of Lemma 1 had pertained only to the particular functions h x = LLx p p ≥ 1 and h x = Lx r r > 0. Professor Einmahl suggested that we go and obtain a version of Lemma 1 for more general h · in the form of an “integral test.” Our new version of Lemma 1 has thus enabled us to present the very general Theorems 3 and 4 rather than the special cases (as in Corollaries 1 and 2) which we originally had presented. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176.

PY - 2008/9

Y1 - 2008/9

N2 - For a sequence of i.i.d. mean 0 random variables {X, Xn; n ≥ 1} with partial sums equation presented, necessary and/or sufficient conditions are provided for {X, Xn; n ≥ 1} to enjoy iterated logarithm type behavior of the form equation presented almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases IEX2 < ∞ and IEX2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an "integral test" involving whether equation presented is finite or infinite where Lx = loge(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).

AB - For a sequence of i.i.d. mean 0 random variables {X, Xn; n ≥ 1} with partial sums equation presented, necessary and/or sufficient conditions are provided for {X, Xn; n ≥ 1} to enjoy iterated logarithm type behavior of the form equation presented almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases IEX2 < ∞ and IEX2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an "integral test" involving whether equation presented is finite or infinite where Lx = loge(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·).

KW - Almost surely

KW - Cluster set

KW - Iterated logarithm type behavior

KW - Law of the iterated logarithm

KW - Slowly varying function

KW - Sums of i.i.d. random variables

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SN - 0736-2994

VL - 26

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EP - 1110

JO - Stochastic Analysis and Applications

JF - Stochastic Analysis and Applications

IS - 5

ER -

A supplement to the Einmahl-Li results on two-sided iterated logarithm type behavior for i.i.d. random variables

Abstract

Bibliographical note

Keywords

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