## The Annals of Probability

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

### On the convergence of scaled random samples

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1065725190

**Digital Object Identifier**

doi:10.1214/aop/1065725190

**Mathematical Reviews number (MathSciNet)**

MR1411503

**Zentralblatt MATH identifier**

0870.60047

**Subjects**

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]

**Keywords**

Scaled sample large deviations regular variation

#### Citation

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