Convergence of weighted sums of tight random elements

Duan Wei and R. L. Taylor

Journal of Multivariate Analysis, 1978, vol. 8, issue 2, 282-294

Abstract: Convergence of weighted sums of tight random elements {Vn} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {ank} is a Toeplitz sequence of real numbers, then [Sigma]k=1[infinity] ankf(Vk) --> 0 in probability for each continuous linear functional f if and only if ||[Sigma]k=1[infinity] ankVk ||--> 0 in probability. When the random elements are independent and max1 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.

Keywords: Random; elements; weighted; sums; tightness; symmetry; Schauder; basis; convergence; in; probability; and; almost; surely; laws; of; large; numbers; Banach; spaces; and; Frechet; spaces (search for similar items in EconPapers)
Date: 1978
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Citations: View citations in EconPapers (4)

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