Moments do not determine tail

Statistics & Probability North-Holland

Letters

15 March

19 (1994) 327-328

Moments do not determine

1994

tail

Min-Te Chao and Wen-Qi Liang Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, ROC Received December Revised May 1993

1992

Abstract: It is an elementary fact that the existence of moments are determined by the tail behavior of the distribution. The converse, however, needs to be discussed. We provide an elementary example that shows random variables with only a few finite moments may have a p.d.f. f(x)lO such that as x -+m, lim inf xbf(x)= 0 for all b > 0. Keywords: Regularly

varying

function

1. Introduction

An elementary fact we learn from probability theory is that whether a certain moment exists depends on the tail behavior of the underlying distribution. For example, all moments are finite for a normal distribution but for the t-distribution, only the first few moments are finite. The basic reason is that the probability density function (p.d.f.) of a t-distribution tends to 0 at ~0with a polynomial rate, and this phenomenon is closely connected with the convergence of the so called p-series CKP. A more refined result in this direction is the following inequality (Chow and Teicher 1978, p. 89): For any random variable X and any r > 0, 2 Pr(lXl

an”‘)

X at cc. The type of question we intend to ask is of the following nature. Suppose X > 0 is a positive random variable with p.d.f. f(x), x > 0. If EX” < 00, then we would have Xb+lf( ,K) + 0

for any 6 G a as x + 03. Now assume X is a random variable such that EXb < CQfor any b < a, but EXb = ~0if b > a, can we say anything more about f(x) at m? Before we go on we would explore our question within the family of distributions RV_, that have tail of the form f(x)

-Cx_PL(x),

where L is a slowly varying function. RV_,, we would have

GEIXI’

n=l

(1.2)

(1.3) For f~

cc

< C Pr(IXI&n’/‘). n=O

(1.1)

Despite its elegance, however, this inequality tells very little about the tail behavior of the p.d.f. of Correspondence ence, Academia Research of ROC.

to: Min-Te Chao, Institute of Statistical Sinica, Taipei 11529, Taiwan, ROC.

partially

supported

by the National

0167-7152/94/$07.00 0 1994 - Elsevier SSDI 0167-7152(93)E0125-D

Science

Science

Sci-

Council

x”f( x) + 00

(1.4)

if b >p - 1. At first sight, it seems natural to suspect that (1.4) may hold in general. After all, (1.2) and (1.4) together give good indication for the thickness of the tail of f. Our question: can we remove the requirement (1.3) that f is asymptotically a regularly varying function at infinity? To make the problem a little harder, we may assume that f(x) > 0 for all x > 0 and f(x) JO.

B.V. Ail rights reserved

327

Volume

19, Number

4

STATISTICS

& PROBABILITY

Define the sequence (a,) by the recursive relation a, = a,_, + 2”n-1, a, = 0 and let =a,3

if x E [a,_,,

a,).

It can be shown that g(x)JO j;;“g(x)

(2.1)

and

dx < co

if and only if b < 2. Hence if we normalize g to form a p.d.f. f of a random variable X, then EXb 0, define p,, = (a~:)‘/~. By construction p,6g(p,) = a if U n-l~
1994

u;g( a,) = ( uJb( a, + 2”m)-3 + 0 for all b > 0, we have lim inf,,,xbg(x) = 0 for all b > 0. We have constructed a p.d.f. f, outside the family of regularly varying functions (Resnick, 1987), that has EXb < co for b < 2 and EXb = cQ for b a 2.

Acknowledgment

We thank the referee for suggesting a simpler version of our original example.

(2.2)

Now if b = 3, (2.2) holds eventually if (YE (0, 1); and if b > 3 then (2.2) holds eventually for all (Y> 0. Hence as x + m, Xbg(X) visits every (Y in (0, 1) or (0, co) infinitely often if b = 3 or b > 3 respectively.

328

15 March

To find the limit infimum of .xbg(x), we take advantage of the discontinuity of g at a,. Since

2. An example

g(x)

LE-ITERS

References Chow, Y.S. and H. Teicher (19781, Probability Theory (Springer, Berlin). Resnick, S.I. (1987), Extreme Values, Regular Vuriafion, and Point Processes (Springer, Berlin).