Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables
Let {Xk} be independent random variables with EXk=0 for all k and let {ank:n≥1, k≥1} be an array of real numbers. In this paper the almost sure convergence of Sn=∑k=1nankXk, n=1,2,…, to a constant is studied under various conditions on the weights {ank} and on the random variables {Xk} using marting...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171279000272 |
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| Summary: | Let {Xk} be independent random variables with EXk=0 for all k and let {ank:n≥1, k≥1} be an array of real numbers. In this paper the almost sure convergence of Sn=∑k=1nankXk, n=1,2,…, to a constant is studied under various conditions on the weights {ank} and on the random variables {Xk} using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces. |
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| ISSN: | 0161-1712 1687-0425 |