## 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 *X**i* are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of *S*_{n}(2) =(∑^{n}_{i}_{=1 }*X*_{i})/(∑^{n}_{j}_{=1}*X*_{j2})^{1/2} as *n*→∞ has density *f*(*t*) = (2*π*)^{−1/2 }exp(−*t*^{2}/2) by the central limit theorem and the law of large numbers. If the tails of *X*_{i} are sufficiently smooth and satisfy *P*(*X*_{i }> *t*) ∼ *rt*^{−α} and *P*(*X*_{i }< −*t*) ∼ *lt*^{−α} as *t*→∞, where 0 < α < 2, *r *> 0, *l *> 0, *S*_{n}(2) still has a limiting distribution *F* even though *X**i* 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 *S*_{n}(*p*) = (∑^{n}_{i}_{=1}*X*_{i})/(∑^{n}_{j}_{=1 }|*X*_{j}|^{p})^{1/p} where *X*_{i} 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

#### Recommended Citation

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.