CLUSTER SETS FOR PARTIAL SUMS AND PARTIAL SUM PROCESSES

• 成果类型: Article
• 署名作者: Einmahl, Uwe; Kuelbs, Jim
• 署名单位: Vrije Universiteit Brussel; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP827
• 发表日期: 2014
• 页码: 1121-1160
• 关键词: iterated logarithm; generalized law; banach-spaces; lil behavior

摘要

Let X, X-1, X-2,... be i.i.d, mean zero random vectors with values in a separable Banach space B, S-n = X-1 +...+ X-n for n >= 1, and assume {c(n) : n >= 1} is a suitably regular sequence of constants. Furthermore, let S((n)) (t), 0 <= t <= 1 be the corresponding linearly interpolated partial sum processes. We study the cluster sets A = C({S-n/c(n)}) and A = C({S((n))(.)/c(n)}). In particular, A and A are shown to be nonrandom, and we derive criteria when elements in B and continuous functions f: [0, 1] -> B belong to A and A, respectively. When B = R-d we refine our clustering criteria to show both A and A are compact, symmetric, and star-like, and also obtain both upper and lower bound sets for A. When the coordinates of X in R-d are independent random variables, we are able to represent A in terms of A and the classical Strassen set K, and, except for degenerate cases, show A is strictly larger than the lower bound set whenever d <= 2. In addition, we show that for any compact, symmetric, star-like subset A of Rd, there exists an X such that the corresponding functional cluster set A is always the lower bound subset. If d = 2, then additional refinements identify A as a subset of {(x(1)g(1), x(2)g(2)) : (x(1), x(2)) is an element of A, g(1), g(2) is an element of K}, which is the functional cluster set obtained when the coordinates are assumed to be independent.