A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables
Let {𝑋,𝑋𝑖;𝑖≥1} be a sequence of independent and identically distributed positive random variables with a continuous distribution function 𝐹, and 𝐹 has a medium tail. Denote 𝑆𝑛=∑𝑛𝑖=1𝑋𝑖,𝑆𝑛∑(𝑎)=𝑛𝑖=1𝑋𝑖𝐼(𝑀𝑛−𝑎<𝑋𝑖≤𝑀𝑛) and 𝑉2𝑛=∑𝑛𝑖=1(𝑋𝑖−𝑋)2, where 𝑀𝑛=max1≤𝑖≤𝑛𝑋𝑖, ∑𝑋=(1/𝑛)𝑛𝑖=1𝑋𝑖, and 𝑎>0 is a fixed const...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2011/181409 |
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| author | Fa-mei Zheng |
| author_facet | Fa-mei Zheng |
| author_sort | Fa-mei Zheng |
| collection | DOAJ |
| description | Let {𝑋,𝑋𝑖;𝑖≥1} be a sequence of independent and identically distributed positive random variables with a continuous distribution function 𝐹, and 𝐹 has a medium tail. Denote 𝑆𝑛=∑𝑛𝑖=1𝑋𝑖,𝑆𝑛∑(𝑎)=𝑛𝑖=1𝑋𝑖𝐼(𝑀𝑛−𝑎<𝑋𝑖≤𝑀𝑛) and 𝑉2𝑛=∑𝑛𝑖=1(𝑋𝑖−𝑋)2, where 𝑀𝑛=max1≤𝑖≤𝑛𝑋𝑖, ∑𝑋=(1/𝑛)𝑛𝑖=1𝑋𝑖, and 𝑎>0 is a fixed constant. Under some suitable conditions, we show that (∏[𝑛𝑡]𝑘=1(𝑇𝑘(𝑎)/𝜇𝑘))𝜇/𝑉𝑛𝑑∫→exp{𝑡0(𝑊(𝑥)/𝑥)𝑑𝑥}𝑖𝑛𝐷[0,1], as 𝑛→∞, where 𝑇𝑘(𝑎)=𝑆𝑘−𝑆𝑘(𝑎) is the trimmed sum and {𝑊(𝑡);𝑡≥0} is a standard Wiener process. |
| format | Article |
| id | doaj-art-9e30bf93de3549138a8d516ab80e248b |
| institution | Kabale University |
| issn | 1687-952X 1687-9538 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
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| series | Journal of Probability and Statistics |
| spelling | doaj-art-9e30bf93de3549138a8d516ab80e248b2025-02-03T01:04:15ZengWileyJournal of Probability and Statistics1687-952X1687-95382011-01-01201110.1155/2011/181409181409A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random VariablesFa-mei Zheng0School of Mathematical Science, Huaiyin Normal University, Huaian 223300, ChinaLet {𝑋,𝑋𝑖;𝑖≥1} be a sequence of independent and identically distributed positive random variables with a continuous distribution function 𝐹, and 𝐹 has a medium tail. Denote 𝑆𝑛=∑𝑛𝑖=1𝑋𝑖,𝑆𝑛∑(𝑎)=𝑛𝑖=1𝑋𝑖𝐼(𝑀𝑛−𝑎<𝑋𝑖≤𝑀𝑛) and 𝑉2𝑛=∑𝑛𝑖=1(𝑋𝑖−𝑋)2, where 𝑀𝑛=max1≤𝑖≤𝑛𝑋𝑖, ∑𝑋=(1/𝑛)𝑛𝑖=1𝑋𝑖, and 𝑎>0 is a fixed constant. Under some suitable conditions, we show that (∏[𝑛𝑡]𝑘=1(𝑇𝑘(𝑎)/𝜇𝑘))𝜇/𝑉𝑛𝑑∫→exp{𝑡0(𝑊(𝑥)/𝑥)𝑑𝑥}𝑖𝑛𝐷[0,1], as 𝑛→∞, where 𝑇𝑘(𝑎)=𝑆𝑘−𝑆𝑘(𝑎) is the trimmed sum and {𝑊(𝑡);𝑡≥0} is a standard Wiener process.http://dx.doi.org/10.1155/2011/181409 |
| spellingShingle | Fa-mei Zheng A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables Journal of Probability and Statistics |
| title | A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables |
| title_full | A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables |
| title_fullStr | A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables |
| title_full_unstemmed | A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables |
| title_short | A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables |
| title_sort | limit theorem for random products of trimmed sums of i i d random variables |
| url | http://dx.doi.org/10.1155/2011/181409 |
| work_keys_str_mv | AT fameizheng alimittheoremforrandomproductsoftrimmedsumsofiidrandomvariables AT fameizheng limittheoremforrandomproductsoftrimmedsumsofiidrandomvariables |