Statistics Papers

Document Type

Journal Article

Date of this Version

1973

Publication Source

The Annals of Probability

Volume

1

Issue

5

Start Page

788

Last Page

809

DOI

10.1214/aop/1176996846

Abstract

If Xi are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sn(2) =(∑ni=1 Xi)/(∑nj=1Xj2)1/2 as n→∞ has density f(t) = (2π)−1/2 exp(−t2/2) by the central limit theorem and the law of large numbers. If the tails of Xi are sufficiently smooth and satisfy P(Xi > t) ∼ rt−α and P(Xi < −t) ∼ lt−α as t→∞, where 0 < α < 2, r > 0, l > 0, Sn(2) still has a limiting distribution F even though Xi has infinite variance. The density f of F depends on α as well as on r/l. We also study the limiting distribution of the more general Sn(p) = (∑ni=1Xi)/(∑nj=1 |Xj|p)1/p where Xi are i.i.d. and in the domain of a stable law G with tails as above. In the cases p = 2 (see (4.21)) and p = 1 (see (3.7)) we obtain exact, computable formulas for f(t) = f(t,α,r/l), and give graphs of f for a number of values of α and r/l. For p = 2, we find that f is always symmetric about zero on (−1,1), even though f is symmetric on (−∞,∞) only when r = l.

Keywords

60F0F, limit theorems, stable laws, maxima of i.i.d characteristic function, domains of attraction

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Date Posted: 27 November 2017

This document has been peer reviewed.