Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions
In the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result genera...
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2025-02-01
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| Series: | Modern Stochastics: Theory and Applications |
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| Online Access: | https://www.vmsta.org/doi/10.15559/25-VMSTA272 |
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| _version_ | 1849687506027544576 |
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| author | Rita Giuliano Milto Hadjikyriakou |
| author_facet | Rita Giuliano Milto Hadjikyriakou |
| author_sort | Rita Giuliano |
| collection | DOAJ |
| description | In the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Lüroth digits and also covers some new cases that have never been studied before. Next, any assumptions concerning the structure of the Oppenheim expansions are dropped and a result concerning trimmed sums is proven when at least two summands are trimmed; combining this latter theorem with the asymptotic behavior of the r-th maximum term of the expansion, a convergence result is obtained for the case in which only the largest summand is deleted from the sum. |
| format | Article |
| id | doaj-art-14eba241c2924fdeb0f2c080dad2f8e7 |
| institution | DOAJ |
| issn | 2351-6046 2351-6054 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | VTeX |
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| series | Modern Stochastics: Theory and Applications |
| spelling | doaj-art-14eba241c2924fdeb0f2c080dad2f8e72025-08-20T03:22:19ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542025-02-0112327328810.15559/25-VMSTA272Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansionsRita Giuliano0Milto Hadjikyriakou1Dipartimento di Matematica Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, ItalySchool of Sciences, University of Central Lancashire, Cyprus campus, CyprusIn the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Lüroth digits and also covers some new cases that have never been studied before. Next, any assumptions concerning the structure of the Oppenheim expansions are dropped and a result concerning trimmed sums is proven when at least two summands are trimmed; combining this latter theorem with the asymptotic behavior of the r-th maximum term of the expansion, a convergence result is obtained for the case in which only the largest summand is deleted from the sum.https://www.vmsta.org/doi/10.15559/25-VMSTA272Oppenheim expansioninfinite expectationlightly trimmed sumlargest summandgood sequenceLüroth series |
| spellingShingle | Rita Giuliano Milto Hadjikyriakou Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions Modern Stochastics: Theory and Applications Oppenheim expansion infinite expectation lightly trimmed sum largest summand good sequence Lüroth series |
| title | Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions |
| title_full | Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions |
| title_fullStr | Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions |
| title_full_unstemmed | Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions |
| title_short | Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions |
| title_sort | strong laws of large numbers for lightly trimmed sums of generalized oppenheim expansions |
| topic | Oppenheim expansion infinite expectation lightly trimmed sum largest summand good sequence Lüroth series |
| url | https://www.vmsta.org/doi/10.15559/25-VMSTA272 |
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