Stochastic convergence of weighted sums in normed linear spaces
Let X be a (real) separable Banach space, let {Vk} be a sequence of random elements in X, and let {ank} be a double array of real numbers such that limn-->[infinity] ank = 0 for all k and [Sigma][infinity]k=1 ank <= 1 for all n. Define Sn = [Sigma]nk=1 ank(Vk - EVk). The convergence of {Sn} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {Vk}. Convergence with probability one for {Sn} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {ank} but without distribution conditions for the random elements {Vk}. Finally, examples of arrays {ank}, spaces, and applications of these results are considered.
Year of publication: |
1975
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Authors: | Taylor, R. L. ; Padgett, W. J. |
Published in: |
Journal of Multivariate Analysis. - Elsevier, ISSN 0047-259X. - Vol. 5.1975, 4, p. 434-450
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Publisher: |
Elsevier |
Keywords: | Random elements Weighted sums Convergence in probability and with probability one Beck-convexity Laws of large numbers Normed linear spaces and Schauder bases |
Saved in:
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