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November, 1983 A Multidimensional CLT for Maxima of Normed Sums
Charles Hagwood, Henry Teicher
Ann. Probab. 11(4): 1048-1050 (November, 1983). DOI: 10.1214/aop/1176993454


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 $d = 1$.


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Charles Hagwood. Henry Teicher. "A Multidimensional CLT for Maxima of Normed Sums." Ann. Probab. 11 (4) 1048 - 1050, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0519.60015
MathSciNet: MR714968
Digital Object Identifier: 10.1214/aop/1176993454

Primary: 60F05
Secondary: 60K05

Keywords: maxima of normed sums , Multivariate CLT , Stopping rules

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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