The Annals of Probability

On the convergence of scaled random samples

Geoffrey Pritchard

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The scaled-sample problem asks the following question: given a distribution on a normed linear space E, when do there exist constants ${\gamma_n} such that $X^{(j)}/\gamma_n}_{j=1}^n$ converges as $n \to \infty$ (in the Hausdorff metric given by the norm) to a fixed set K? (Here ${X^{(j)}}$ are i.i.d. with the given distribution). The main result presented here relates the convergence of scaled samples to a large deviation principle for single observations, thereby achieving a dimension-free description of the problem.

Article information

Ann. Probab., Volume 24, Number 3 (1996), 1490-1506.

First available in Project Euclid: 9 October 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60F15: Strong theorems 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Scaled sample large deviations regular variation


Pritchard, Geoffrey. On the convergence of scaled random samples. Ann. Probab. 24 (1996), no. 3, 1490--1506. doi:10.1214/aop/1065725190.

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