Growth Rates of Moment Sequences

• Activity Number: 585
• Type: Contributed
• Date/Time: Wednesday, August 4, 2010 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): William L. Harkness*+
• Companies: Penn State
• Keywords: Ratios of Moments ; Growth rates ; regularly varying functions

Abstract

General properties of growth rates of moment sequences of nonnegative random variables are presented. Then asymptotic results on moment sequences are derived for two classes of distribution functions. Explicitly, let g be a monotone increasing twice differentiable regularly varying function at infinity with g(+8)=+8 and index of variation ? = 0. Define the distribution function F by -ln[1-F(x)] = g-inverse(x). Then an explicit expression for the nth moment is given in terms of g and it is shown that the ratio of the (n+1)st and nth moments is asymptotically equal to g(tn), where tn satisfies the equation ng'(tn)/g(tn)=1. A second class of distribution functions is defined by setting -ln[1-F(x)] = g-inverse(ln x), where now ? = 1, and similar results are obtained in this case. Finally, examples are given to illustrate the possible asymptotic growth rates of moments.