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International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 365-374

A generalization of the global limit theorems of R. P. Agnew

Department of Statistics, University of Florida, Gainesville 32611, Florida, USA

Received 21 May 1987; Revised 24 August 1987

Copyright © 1988 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For distribution functions {Fn,n0}, the relationship between the weak convergence of Fn to F0 and the convergence of Rϕ(|FnF0|)dx to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew wherein ϕ(t)=|t|r. The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line.