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T.-C. Hu, M. Ordonez Cabrera, S. H. Sung, and A.I. Volodin

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Complete convergence of weighted sums in Banach spaces
and the bootstrap mean

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*## (Lobachevskii Journal of Mathematics,

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Vol.10, pp.17-26)

Let $\{X_{ni}, 1\le i\le k_n, n\ge 1\}$ be an array of rowwise
independent
random elements taking values in a real separable Banach space, and
$\{a_{ni}, 1\le i \le k_n, n\ge 1\}$ an array of constants. Under some
conditions
of Chung [7] and Hu and Taylor [10] types for the arrays, and using a
theorem of
Hu et al. [9], the equivalence amongst various kinds of convergence
of $\dis\sum_{i=1}^{k_n}a_{ni}X_{ni}$ to zero is obtained. It leads to
an unified vision of recent results in the literature. The authors use
the main result in the paper in order to obtain the strong consistency
of the bootstrapped mean of random elements in a Banach space from its
weak consistency.

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