Darling-Erdos theorem for self-normalized sums

Miklós Csörgo; Barbara Szyszkowicz; Qiying Wang

doi:10.1214/aop/1048516532

Darling-Erdos theorem for self-normalized sums

Miklós Csörgo^*, Barbara Szyszkowicz, Qiying Wang

^*Corresponding author for this work

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

25 Citations (Scopus)

Abstract

Let X, X₁, X₂,... be i.i.d. nondegenerate random variables, S_n = ∑_j=1ⁿ X_j and V_n² = ∑_j=1ⁿ. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max_1≤k≤n S_k/V_k, and the law of the iterated logarithm for self-normalized sums, S_n/V_n, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling-Erdos-type theorem as well as an Erdos-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.

Original language	English
Pages (from-to)	676-692
Number of pages	17
Journal	Annals of Probability
Volume	31
Issue number	2
DOIs	https://doi.org/10.1214/aop/1048516532
Publication status	Published - Apr 2003

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title = "Darling-Erdos theorem for self-normalized sums",

abstract = "Let X, X1, X2,... be i.i.d. nondegenerate random variables, Sn = ∑j=1n Xj and Vn2 = ∑j=1n. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤k≤n Sk/Vk, and the law of the iterated logarithm for self-normalized sums, Sn/Vn, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling-Erdos-type theorem as well as an Erdos-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.",

keywords = "Darling-Erdos theorem, Erdos-Feller-Kolmogorov-Petrovski test, Self-normalized sums",

author = "Mikl{\'o}s Cs{\"o}rgo and Barbara Szyszkowicz and Qiying Wang",

year = "2003",

month = apr,

doi = "10.1214/aop/1048516532",

language = "English",

volume = "31",

pages = "676--692",

journal = "Annals of Probability",

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publisher = "Institute of Mathematical Statistics",

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TY - JOUR

T1 - Darling-Erdos theorem for self-normalized sums

AU - Csörgo, Miklós

AU - Szyszkowicz, Barbara

AU - Wang, Qiying

PY - 2003/4

Y1 - 2003/4

N2 - Let X, X1, X2,... be i.i.d. nondegenerate random variables, Sn = ∑j=1n Xj and Vn2 = ∑j=1n. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤k≤n Sk/Vk, and the law of the iterated logarithm for self-normalized sums, Sn/Vn, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling-Erdos-type theorem as well as an Erdos-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.

AB - Let X, X1, X2,... be i.i.d. nondegenerate random variables, Sn = ∑j=1n Xj and Vn2 = ∑j=1n. We investigate the asymptotic behavior in distribution of the maximum of self-normalized sums, max1≤k≤n Sk/Vk, and the law of the iterated logarithm for self-normalized sums, Sn/Vn, when X belongs to the domain of attraction of the normal law. In this context, we establish a Darling-Erdos-type theorem as well as an Erdos-Feller-Kolmogorov-Petrovski-type test for self-normalized sums.

KW - Darling-Erdos theorem

KW - Erdos-Feller-Kolmogorov-Petrovski test

KW - Self-normalized sums

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U2 - 10.1214/aop/1048516532

DO - 10.1214/aop/1048516532

M3 - Article

SN - 0091-1798

VL - 31

SP - 676

EP - 692

JO - Annals of Probability

JF - Annals of Probability

IS - 2

ER -

Darling-Erdos theorem for self-normalized sums

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