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April 2003 Darling--Erdős theorem for self-normalized sums
Miklós Csörgő, Barbara Szyszkowicz, Qiying Wang
Ann. Probab. 31(2): 676-692 (April 2003). DOI: 10.1214/aop/1048516532

Abstract

Let $X,\, X_1,\, X_2,\ldots$ be i.i.d. nondegenerate random variables, $S_n=\sum_{j=1}^nX_j$ and $V_n^2=\sum_{j=1}^nX_j^2$. We investigate the asymptotic \vspace*{1pt} behavior in distribution of the maximum of self-normalized sums, $\max_{1\le k\le n}S_k/V_k$, 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--Erdős-type theorem as well as an Erdős--Feller--Kolmogorov--Petrovski-type test for self-normalized sums.

Citation

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Miklós Csörgő. Barbara Szyszkowicz. Qiying Wang. "Darling--Erdős theorem for self-normalized sums." Ann. Probab. 31 (2) 676 - 692, April 2003. https://doi.org/10.1214/aop/1048516532

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1035.60017
MathSciNet: MR1964945
Digital Object Identifier: 10.1214/aop/1048516532

Subjects:
Primary: 60F05 , 60F15
Secondary: 62E20

Keywords: Darling--Erdős theorem , Erdős--Feller--Kolmogorov--Petrovski test , self-normalized sums

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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