 |
|
|
| | | | | |
|
|
|
|
|
Some LIL type results on the partial sums and trimmed sums with multidimensional indices
|
Wei-Dong Liu, Zhejiang University
Zheng-Yan Lin, Zhejiang University
|
Abstract
Let {X, Xn;
n ∈ Nd} be a field of i.i.d. random variables indexed by
d-tuples of positive integers and let
Sn=∑k≤nXk. We prove
some strong limit theorems for Sn. Also, when d≥ 2
and h(n)
satisfies some conditions, we show that there are no LIL type results for
Sn/√|n|h(n)
|
Full text: PDF
Pages: 221-233
Published on: July 14, 2007
|
Bibliography
- Einmahl, Uwe; Li, Deli. Some results on two-sided LIL behavior.
Ann. Probab. 33 (2005), no. 4, 1601--1624. MR2150200 (2006b:60049)
- Einmahl, Uwe; Mason, David M. Gaussian approximation of local empirical processes indexed by
functions.
Probab. Theory Related Fields 107 (1997), no. 3, 283--311. MR1440134 (98d:60060)
- Feller, William. An extension of the law of the iterated logarithm to variables without
variance.
J. Math. Mech. 18 1968/1969 343--355. MR0233399 (38 #1721)
- Gut, Allan. Marcinkiewicz laws and convergence rates in the law of large numbers
for random variables with multidimensional indices.
Ann. Probability 6 (1978), no. 3, 469--482. MR0494431 (58 #13296)
- Gut, Allan. Convergence rates for probabilities of moderate deviations for sums of
random variables with multidimensional indices.
Ann. Probab. 8 (1980), no. 2, 298--313. MR0566595 (81c:60035)
- Klass, M. J. Toward a universal law of the iterated logarithm. I.
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 2, 165--178. MR0415742 (54 #3822)
- Klass, Michael J. Toward a universal law of the iterated logarithm. II.
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 39 (1977), no. 2, 151--165. MR0448502 (56 #6808)
- Li, De Li. Some results on strong limit theorems for multidimensional
indices.
(Chinese) Acta Math. Sinica 33 (1990), no. 3, 402--413. MR1070224 (92c:60041)
- Li, Deli; Tomkins, R. J. Compact laws of the iterated logarithm for $B$-valued random
variables with two-dimensional indices.
J. Theoret. Probab. 11 (1998), no. 2, 443--459. MR1622581 (99e:60019)
- Smythe, R. T. Strong laws of large numbers for $r$-dimensional arrays of random
variables.
Ann. Probability 1 (1973), no. 1, 164--170. MR0346881 (49 #11602)
- Wichura, Michael J. Some Strassen-type laws of the iterated logarithm for multiparameter
stochastic processes with independent increments.
Ann. Probability 1 (1973), 272--296. MR0394894 (52 #15693)
- Zhang, Li Xin. Strong approximation theorems for sums of random variables when extreme
terms are excluded.
Acta Math. Sin. (Engl. Ser.) 18 (2002), no. 2, 311--326. MR1910967 (2003e:60065)
|
|
|
|
|
|
|
|
| | | | |
Electronic Communications in Probability. ISSN: 1083-589X |
|
Context