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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| Format: | Article |
| Language: | English |
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Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171279000272 |
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| author | W. J. Padgett R. L. Taylor |
| author_facet | W. J. Padgett R. L. Taylor |
| author_sort | W. J. Padgett |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f586460efe04444a85f21284a17e507c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1979-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-f586460efe04444a85f21284a17e507c2025-02-03T05:52:17ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251979-01-012230932310.1155/S0161171279000272Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variablesW. J. Padgett0R. L. Taylor1Department of Mathematics, Computer Science, and Statistics, University of South Carolina, Columbia, Columbia 29208, South Carolina, USADepartment of Mathematics, Computer Science, and Statistics, University of South Carolina, Columbia, Columbia 29208, South Carolina, USALet {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.http://dx.doi.org/10.1155/S0161171279000272weighted sumsstrong law of large numbersalmost sure convergencegenerlized Gaussian random variablesrandom elements in Banach spaceSchauder basis. |
| spellingShingle | W. J. Padgett R. L. Taylor Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables International Journal of Mathematics and Mathematical Sciences weighted sums strong law of large numbers almost sure convergence generlized Gaussian random variables random elements in Banach space Schauder basis. |
| title | Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables |
| title_full | Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables |
| title_fullStr | Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables |
| title_full_unstemmed | Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables |
| title_short | Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables |
| title_sort | convergence of weighted sums of independent random variables and an extension to banach space valued random variables |
| topic | weighted sums strong law of large numbers almost sure convergence generlized Gaussian random variables random elements in Banach space Schauder basis. |
| url | http://dx.doi.org/10.1155/S0161171279000272 |
| work_keys_str_mv | AT wjpadgett convergenceofweightedsumsofindependentrandomvariablesandanextensiontobanachspacevaluedrandomvariables AT rltaylor convergenceofweightedsumsofindependentrandomvariablesandanextensiontobanachspacevaluedrandomvariables |