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A strong law for $ B$-valued arrays

Authors: De Li Li, M. Bhaskara Rao and R. J. Tomkins
Journal: Proc. Amer. Math. Soc. 123 (1995), 3205-3212
MSC: Primary 60B12; Secondary 60F15
MathSciNet review: 1291783
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Abstract: Let $ (B,\left\Vert \bullet \right\Vert)$ be a real separable Banach space and $ \{ {X_{n,k}};n \geq 1,1 \leq k \leq n\} $ a triangular array of iid B-valued random variables. Set $ S(n) = \sum\nolimits_{k = 1}^n {{X_{n,k}},n \geq 1} $, and $ {\operatorname{Log}}\,t = \log \max \{ e,t\} ,t \in \Re $. In this paper, we characterize the limit behavior of $ S(n)/\sqrt {2n\,{\operatorname{Log}}\,n} ,n \geq 1$. As an application of our result, we resolve an open problem posed by Hu and Weber (1992). The case of row-wise independent arrays is also dealt with.

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  • [1] E. Berger, Majorization, exponential inequalities, and almost sure behavior of vector valued random variables, Ann. Probab. 19 (1991), 1206-1226. MR 1112413 (92j:60006)
  • [2] X. Chen, Probabilities of moderate deviations for independent random vectors in a Banach space, Chinese J. Appl. Probab. Statist. 7 (1991), 24-32. MR 1204537 (93m:60020)
  • [3] A. de Acosta, Moderate deviations and associated Laplace approximations for sums of independent random variables, Trans. Amer. Math. Soc. 329 (1992), 357-375. MR 1046015 (92e:60053)
  • [4] T. C. Hu, F. Móricz, and R. L. Taylor, Strong Laws of Large Numbers for arrays of rowwise independent random variables, Acta Math. Hungar. 54 (1989), 153-162. MR 1015785 (91f:60061)
  • [5] T. C. Hu and N. C. Weber, On the rate of convergence in the Strong Law of Large Numbers for arrays, Bull. Austral. Math. Soc. 45 (1992), 479-482. MR 1165154 (93d:60044)
  • [6] D. Kaffes and M. Bhaskara Rao, Weak consistency of least-squares estimators in linear models, J. Multivariate Anal. 12 (1982), 186-198. MR 661558 (83m:62117)
  • [7] J. Kuelbs, A strong convergence theorem for Banach space valued random variables, Ann. Probab. 4 (1976), 744-771. MR 0420771 (54:8783)
  • [8] T. L. Lai, Limit theorems for delayed sums, Ann. Probab. 2 (1974), 432-440. MR 0356193 (50:8664)
  • [9] M. Ledoux, Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi, Ann. Inst. Henri Poincaré 28 (1992), 267-280. MR 1162575 (93k:60017)
  • [10] D. Li, Convergence rates of law of iterated logarithm for B-valued random variables, Sci. China Ser. A 34 (1991), 395-404. MR 1119489 (92j:60007)
  • [11] D. Li and X. C. Wang, Convergence rates for probabilities of moderate deviations for sums of B-valued random variables, Proceeding of the Special Year in Probability and Statistics (Tianjin, China, August 1988-May 1989), World Scientific, River Edge, NJ, 1992, pp. 100-110. MR 1176548 (93j:60006)
  • [12] R. J. Tomkins, On the law of the iterated logarithm for double sequence of random variables, Z. Wahrsch. Verw. Gebiete 30 (1974), 303-314. MR 0405555 (53:9348)

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Keywords: Almost sure limit, Banach space, cluster set, the Law of the Iterated Logarithm, Strong Law of Large Numbers
Article copyright: © Copyright 1995 American Mathematical Society

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