## The Annals of Probability

### Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law

Jean Bertoin

#### Abstract

Let $\xi, \xi_1, \dots$ be i.i.d. real-valued random variables and $S_n = \xi_1 + \dots + \xi_n$. In the case when the distribution of $\xi$ is close to a stable $(\alpha)$ law for some $\alpha \epsilon (0, 1) \bigcup (1, 2)$, we investigate the asymptotic behavior in distribution of the maximum of normalized sums, $\max_{k=1,\dots,n} k^{-1/\alpha}S_k$. This completes the Darling-Erdös limit theorem for the case $\alpha = 2$.

#### Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 832-852.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855652

Digital Object Identifier
doi:10.1214/aop/1022855652

Mathematical Reviews number (MathSciNet)
MR1626527

Zentralblatt MATH identifier
0943.60041

Subjects
Primary: 60J30
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes

#### Citation

Bertoin, Jean. Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law. Ann. Probab. 26 (1998), no. 2, 832--852. doi:10.1214/aop/1022855652. https://projecteuclid.org/euclid.aop/1022855652

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