Open Access
Translator Disclaimer
April, 1978 A Weak Invariance Principle with Applications to Domains of Attraction
Gordon Simons, William Stout
Ann. Probab. 6(2): 294-315 (April, 1978). DOI: 10.1214/aop/1176995574


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.


Download Citation

Gordon Simons. William Stout. "A Weak Invariance Principle with Applications to Domains of Attraction." Ann. Probab. 6 (2) 294 - 315, April, 1978.


Published: April, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0376.60027
MathSciNet: MR471037
Digital Object Identifier: 10.1214/aop/1176995574

Primary: 60F05
Secondary: 60G50

Keywords: Central limit problem , domain of attraction , domain of normal attraction , domain of partial attraction , invariance principle , quantile function , slowly varying function , stable distribution

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 2 • April, 1978
Back to Top