Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
7547785 | Statistics & Probability Letters | 2018 | 14 Pages |
Abstract
Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,â¦) and Sn(a)=âi=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bnââ such that W¯n(a)=bnâ1Sn(a)âp1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is â1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1âF(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infnââW¯n(a)=1 and lim supnââW¯n(a)=â.
Related Topics
Physical Sciences and Engineering
Mathematics
Statistics and Probability
Authors
André Adler, Anthony G. Pakes,