Abstract
For a sequence of i.i.d. mean 0 random variables {X, Xn ; n ≥ 1} with weighted partial sums Sn (X, w ({dot operator})) = ∑k = 1n w (frac(k, n)) Xk, n ≥ 1 where w (t), 0 ≤ t ≤ 1 is a Lipschitz function of order 1 with {norm of matrix} w ({dot operator}) {norm of matrix}2 = sqrt(∫01 w2 (t) d t) > 0, necessary and sufficient conditions are provided for X to enjoy iterated logarithm type behavior of the form 0 < lim supn → ∞ | Sn (X, w ({dot operator})) | / sqrt(n h (n)) < ∞ almost surely where h ({dot operator}) is a positive, nondecreasing function which is slowly varying at infinity. Some corollaries are presented for particular choices of h ({dot operator}).
| Original language | English (US) |
|---|---|
| Pages (from-to) | 643-651 |
| Number of pages | 9 |
| Journal | Statistics and Probability Letters |
| Volume | 79 |
| Issue number | 5 |
| DOIs | |
| State | Published - Mar 1 2009 |
Bibliographical note
Funding Information:The authors are grateful to the referee for carefully reading the manuscript and for pointing out to them obscureness and gaps in some of the arguments in the original version. The authors are also grateful to Professor Uwe Einmahl for his interest in their work and for his comments on a preliminary version of Theorem 1 . The perceptive comments of the referee and Professor Einmahl enabled the authors to substantially improve the paper. The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada and the research of Yongcheng Qi was partially supported by NSF Grant DMS-0604176.