Abstract
Let {Vk∶k≥1} be a sequence of random elements in a real separable normed linear space X, and let {ank∶n≥1, k≥1} be an array of real numbers. Several theorems are given which provide conditions for the convergence with probability one of \(s_n = \sum\nolimits_{k = 1}^n {a_{nk} V_k } \)to the zero element of X. One result states that if X is B-convex and if the random elements are independent with expected values zero and uniformly bounded rth moments for some r>1, then, under a given set of conditions on {ank}, Sn→0 in X with probability one.
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Padgett, W.J., Taylor, R.L. (1976). Almost sure convergence of weighted sums of random elements in Banach spaces. In: Beck, A. (eds) Probability in Banach Spaces. Lecture Notes in Mathematics, vol 526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082353
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DOI: https://doi.org/10.1007/BFb0082353
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