INFINITE LIMITS AND INFINITE LIMIT POINTS OF RANDOM-WALKS AND TRIMMED SUMS
成果类型：
Article
署名作者：
KESTEN, H; MALLER, RA
署名单位：
University of Western Australia
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176988609
发表日期：
1994
页码：
1473-1513
关键词：
摘要：
We consider infinite limit points (in probability) for sums and lightly trimmed sums of i.i.d, random variables normalized by a nonstochastic sequence. More specifically, let X(1),X(2),... be independent random variables with common distribution F. Let M(n)((r))) be the rth largest among X(1),...,X,,; also let; X(n)((r)) be the observation with the rth largest absolute value among X(1),...,X(n). Set S-n = Sigma(1)(n)X(i), ((r))S-n = S-n - M(n)((1)) - ... - M(n)((r)) and ((r))(S) over tilde(n) = S-n - X(n)((1)) -...-X(n)((r)) (((0))S-n = ((0))(S) over tilde(n) = S-n). We find simple criteria in terms of F for (r)S-n/B-n --> p +/- infinity (i.e., ((r))S-n/B-n tends to infinity or to -infinity in probability) or ((r))(S) over tilde(n)/B-n --> p +/- infinity when r = 0, 1,.... Here B-n up arrow infinity may be given in advance, or its existence may be investigated. In particular, we find a necessary and sufficient condition for ((r))S-n/n --> p infinity. Some equivalences for the divergence of \((r))(S) over tilde(n)\/\X(n)((r))\, or of ((r))S-n/(X(-))(n)((s)), where (X(-))(n)((s)) is the sth largest of the negative parts of the X(i), and for the convergence P{S-n > 0} --> 1, as n --> infinity, are also proven. In some cases we treat divergence along a subsequence as well, and one such result provides an equivalence for a generalized iterated logarithm law due to Pruitt.