TY - JOUR
T1 - Weak and one-sided strong laws for random variables with infinite mean
AU - Adler, André
AU - Pakes, Anthony G.
PY - 2018/11/1
Y1 - 2018/11/1
N2 - Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1 naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn −1Sn(a)→p1, 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 −1. 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(1−F(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 infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.
AB - Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1 naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn −1Sn(a)→p1, 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 −1. 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(1−F(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 infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.
KW - Regular variation
KW - Relative stability
KW - Weighted laws of large numbers
UR - http://www.scopus.com/inward/record.url?scp=85049481471&partnerID=8YFLogxK
U2 - 10.1016/j.spl.2018.06.008
DO - 10.1016/j.spl.2018.06.008
M3 - Article
AN - SCOPUS:85049481471
VL - 142
SP - 8
EP - 16
JO - Statistics & Probability Letters
JF - Statistics & Probability Letters
SN - 0167-7152
ER -