Open Access
July, 1988 Characteristics of Normal Samples
Victor Goodman
Ann. Probab. 16(3): 1281-1290 (July, 1988). DOI: 10.1214/aop/1176991690


A "law of large numbers" for the maximum of i.i.d. univariate normal random variables is extended to a general multivariate case. Let $\mathbf{Z}_i$ denote i.i.d. Banach space valued random variables with a centered Gaussian distribution. Let $\mathbf{K}$ denote the unit ball of the reproducing kernel Hilbert space. Then with probability 1, the maximum distance from the sample points $\mathbf{Z}_1, \mathbf{Z}_2,\ldots, \mathbf{Z}_n$ to the set $\sqrt{2 \log n} \mathbf{K}$ approaches zero. In addition, the sample forms epsilon nets for this set as $n$ tends to infinity.


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Victor Goodman. "Characteristics of Normal Samples." Ann. Probab. 16 (3) 1281 - 1290, July, 1988.


Published: July, 1988
First available in Project Euclid: 19 April 2007

zbMATH: 0712.60006
MathSciNet: MR942768
Digital Object Identifier: 10.1214/aop/1176991690

Primary: 60B11
Secondary: 60B12 , 60D05 , 60F10 , 60F20 , 60G15

Keywords: cluster set , Gaussian processes , i.i.d. samples , reproducing kernels

Rights: Copyright © 1988 Institute of Mathematical Statistics

Vol.16 • No. 3 • July, 1988
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