## The Annals of Probability

### Darling--Erdős theorem for self-normalized sums

#### 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.

#### Article information

Source
Ann. Probab., Volume 31, Number 2 (2003), 676-692.

Dates
First available in Project Euclid: 24 March 2003

https://projecteuclid.org/euclid.aop/1048516532

Digital Object Identifier
doi:10.1214/aop/1048516532

Mathematical Reviews number (MathSciNet)
MR1964945

Zentralblatt MATH identifier
1035.60017

Subjects
Primary: 60F05: Central limit and other weak theorems 60F15: Strong theorems
Secondary: 62E20: Asymptotic distribution theory

#### Citation

Csörgő, Miklós; Szyszkowicz, Barbara; Wang, Qiying. Darling--Erdős theorem for self-normalized sums. Ann. Probab. 31 (2003), no. 2, 676--692. doi:10.1214/aop/1048516532. https://projecteuclid.org/euclid.aop/1048516532

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• CANBERRA, ACT 0200 AUSTRALIA E-MAIL: qiying@wintermute.anu.edu.au