Open Access
Translator Disclaimer
January, 1986 Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function
Miklos Csorgo, Sandor Csorgo, Lajos Horvath, David M. Mason
Ann. Probab. 14(1): 86-118 (January, 1986). DOI: 10.1214/aop/1176992618


We prove general invariance principles for integral functions of the empirical process. As corollaries we derive probabilistic proofs of the sufficiency criteria for a distribution to belong to the domain of attraction of the normal and stable laws with index $0 < \alpha < 2$. In the process we obtain equivalent statements of these criteria in terms of the tail behaviour of the underlying quantile function. We also give a representation of any stable random variable with index $0 < \alpha < 2$ in terms of a linear combination of two independent and identically distributed Poisson integrals. The role of a fixed number of extreme terms is exactly determined.


Download Citation

Miklos Csorgo. Sandor Csorgo. Lajos Horvath. David M. Mason. "Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function." Ann. Probab. 14 (1) 86 - 118, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0589.60030
MathSciNet: MR815961
Digital Object Identifier: 10.1214/aop/1176992618

Primary: 60F17
Secondary: 60E07 , 60F05

Keywords: Empirical distribution function , Integral functionals , normal convergence criteria , Poisson integrals , quantiles , stable convergence criteria

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 1 • January, 1986
Back to Top