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 $\{X, X_{{n}}; n\in\mathbb{N}^{d}\}$ be a field of i.i.d. random variables indexed by $d$-tuples of positive integers and let $S_{{n}}=\sum_{{k}\leq{n}}X_{{k}}$. We prove some strong limit theorems for $S_{{n}}$. Also, when $d\geq 2$ and $h({n})$ satisfies some conditions, we show that there are no LIL type results for $S_{{n}}/\sqrt{|{n}|h({n})}$.

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Pages: 221-233

Publication Date: July 2, 2007

DOI: 10.1214/ECP.v12-1286

References

  1. Einmahl, Uwe; Li, Deli. Some results on two-sided LIL behavior. Ann. Probab. 33 (2005), no. 4, 1601--1624. MR2150200 (2006b:60049)
  2. 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)
  3. 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)
  4. 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)
  5. 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)
  6. 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)
  7. 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)
  8. 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)
  9. 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)
  10. 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)
  11. 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)
  12. 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)
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