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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
DOI: https://doi.org/10.1090/S0002-9939-1995-1291783-8
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1291783-8
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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