Date of this Version
The Annals of Probability
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.
60F0F, limit theorems, stable laws, maxima of i.i.d characteristic function, domains of attraction
Logan, B. F., Mallows, C. L., Rice, S. O., & Shepp, L. A. (1973). Limit Distributions of Self-Normalized Sums. The Annals of Probability, 1 (5), 788-809. http://dx.doi.org/10.1214/aop/1176996846
Date Posted: 27 November 2017
This document has been peer reviewed.