Darling-Erdős theorems for normalized sums of i. i. d. variables close to a stable law
type: Item
instance of: scholarly article
title: Darling-Erdős theorems for normalized sums of i. i. d. variables close to a stable law
zbMATH Open document ID: 0943.60041
author: Jean Bertoin
published in: The Annals of Probability
publication date: 2000-09-04T00:00:00Z
review text: Let \(\{S_n\), \(n\geq 1\}\) be partial sums of i.i.d. random variables with mean 0 and finite moment of order three. In the 50's Darling and Erdős proved that \(\max_{k\leq n} k^{-1/2} S_k\), suitably normalized, converges in distribution to the Gumbel distribution. The assumption on third order moments has since then been weakened. In this paper corresponding results are proved in the case when the summands belong to the domain of attraction of a strictly stable law with index \(\alpha\in (0,1)\cup (1,2)\). An important tool is analog for extremes of spectrally negative Ornstein-Uhlenbeck processes.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
zbMATH DE Number: 1355255
zbMATH Keywords: convergence in distribution
zbMATH Keywords: Gumbel distribution
zbMATH Keywords: strictly stable law
zbMATH Keywords: spectrally negative Ornstein-Uhlenbeck processes
reviewed by: Allan Gut
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022855652
OpenAlex ID: W1527493561
cites work: On extremal theory for stationary processes
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cites work: Fluctuation theory in continuous time
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cites work: A Delicate Law of the Iterated Logarithm for Non-Decreasing Stable Processes
cites work: A limit theorem for the maximum of normalized sums of independent random variables
cites work: One-sided local large deviation and renewal theorems in the case of infinite mean
cites work: The Darling-Erdős theorem for sums of i.i.d. random variables
cites work: Darling-Erdős theorems for martingales
cites work: A Limit Theoerm for Random Variables with Infinite Moments
cites work: Q5624436
cites work: Large deviations of sums of independent random variables
cites work: The growth of random walks and Levy processes
cites work: Q3771297
cites work: Extremes of moving averages of stable processes
cites work: Extreme value theory for moving average processes
cites work: Maxima and exceedances of stationary Markov chains
cites work: Q4301585
cites work: Extension of the Darling and Erdős theorem on the maximum of normalized sums
cites work: Refined Large Deviation Limit Theorems
cites work: Q3717907
DOI: 10.1214/AOP/1022855652
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description: scientific article; zbMATH DE number 1355255