The Annals of Probability

A Multidimensional CLT for Maxima of Normed Sums

Charles Hagwood and Henry Teicher

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Abstract

It is shown that if $S_{k,j} = \sum^k_{i = 1} X_{ij}, 1 \leq j \leq d, k \geq 1$ where $(X_{i1}, \cdots, X_{id}), i \geq 1$ are i.i.d. random vectors with positive mean vector $(\mu_1, \cdots, \mu_d)$ and finite covariance matrix $\Sigma$, then for any choice of $\alpha_j$ in $\lbrack 0, 1), 1 \leq j \leq d$ the random vector whose $j$th component is $n^{\alpha_j - 1/2} (\max_{1 \leq k \leq n}S_{k,j}/k^{\alpha_j} - \mu_jn^{1 - \alpha_j})$ converges in law to a multinormal distribution with mean vector zero and covariance matrix $\Sigma$, thereby extending a result of Teicher when $d = 1$.

Article information

Source
Ann. Probab., Volume 11, Number 4 (1983), 1048-1050.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993454

Digital Object Identifier
doi:10.1214/aop/1176993454

Mathematical Reviews number (MathSciNet)
MR714968

Zentralblatt MATH identifier
0519.60015

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K05: Renewal theory

Keywords
Multivariate CLT maxima of normed sums stopping rules

Citation

Hagwood, Charles; Teicher, Henry. A Multidimensional CLT for Maxima of Normed Sums. Ann. Probab. 11 (1983), no. 4, 1048--1050. doi:10.1214/aop/1176993454. https://projecteuclid.org/euclid.aop/1176993454


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