The Annals of Probability

A Weak Invariance Principle with Applications to Domains of Attraction

Gordon Simons and William Stout

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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