The Annals of Probability

Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function

Miklos Csorgo, Sandor Csorgo, Lajos Horvath, and David M. Mason

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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.

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Ann. Probab., Volume 14, Number 1 (1986), 86-118.

First available in Project Euclid: 19 April 2007

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Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60E07: Infinitely divisible distributions; stable distributions

Integral functionals empirical distribution function normal convergence criteria stable convergence criteria quantiles Poisson integrals


Csorgo, Miklos; Csorgo, Sandor; Horvath, Lajos; Mason, David M. Normal and Stable Convergence of Integral Functions of the Empirical Distribution Function. Ann. Probab. 14 (1986), no. 1, 86--118. doi:10.1214/aop/1176992618.

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