The Annals of Probability

A Weak Invariance Principle with Applications to Domains of Attraction

Gordon Simons and William Stout

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Abstract

An elementary probabilistic argument is given which establishes a "weak invariance principle" which in turn implies the sufficiency of the classical assumptions associated with the weak convergence of normed sums to stable laws. The argument, which uses quantile functions (the inverses of distribution functions), exploits the fact that two random variables $X = F^{-1}(U)$ and $Y = G^{-1}(U)$ are, in a useful sense, close together when $F$ and $G$ are, in a certain sense, close together. Here $U$ denotes a uniform variable on (0, 1). By-products of the research are two alternative characterizations for a random variable being in the domain of partial attraction to a normal law and some results concerning the study of domains of partial attraction.

Article information

Source
Ann. Probab., Volume 6, Number 2 (1978), 294-315.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995574

Digital Object Identifier
doi:10.1214/aop/1176995574

Mathematical Reviews number (MathSciNet)
MR471037

Zentralblatt MATH identifier
0376.60027

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks

Keywords
Invariance principle domain of attraction domain of partial attraction stable distribution central limit problem domain of normal attraction slowly varying function quantile function

Citation

Simons, Gordon; Stout, William. A Weak Invariance Principle with Applications to Domains of Attraction. Ann. Probab. 6 (1978), no. 2, 294--315. doi:10.1214/aop/1176995574. https://projecteuclid.org/euclid.aop/1176995574


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