The inverse moment for widely orthant dependent random variables
Abstract In this paper, we investigate approximations of the inverse moment model by widely orthant dependent (WOD) random variables. Let { Z n , n ≥ 1 } $\{Z_{n},n\geq1\}$ be a sequence of nonnegative WOD random variables, and { w n i , 1 ≤ i ≤ n , n ≥ 1 } $\{w_{ni},1\leq i\leq n,n\geq 1\}$ be a tr...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-06-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-016-1099-8 |
| _version_ | 1811268838371622912 |
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| author | Xiaoqin Li Xi Liu Wenzhi Yang Shuhe Hu |
| author_facet | Xiaoqin Li Xi Liu Wenzhi Yang Shuhe Hu |
| author_sort | Xiaoqin Li |
| collection | DOAJ |
| description | Abstract In this paper, we investigate approximations of the inverse moment model by widely orthant dependent (WOD) random variables. Let { Z n , n ≥ 1 } $\{Z_{n},n\geq1\}$ be a sequence of nonnegative WOD random variables, and { w n i , 1 ≤ i ≤ n , n ≥ 1 } $\{w_{ni},1\leq i\leq n,n\geq 1\}$ be a triangular array of nonnegative nonrandom weights. If the first moment is finite, then E ( a + ∑ i = 1 n w n i Z i ) − α ∼ ( a + ∑ i = 1 n w n i E Z i ) − α $E(a+ \sum_{i=1}^{n}w_{ni}Z_{i})^{-\alpha}\sim (a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{-\alpha}$ for all constants a > 0 $a>0$ and α > 0 $\alpha>0$ . If the rth moment ( r > 2 $r>2$ ) is finite, then the convergence rate is presented as E ( a + ∑ i = 1 n w n i Z i ) − α ( a + ∑ i = 1 n w n i E Z i ) − α − 1 = O ( 1 ( a + ∑ i = 1 n w n i E Z i ) 1 − 2 β / r ) $\frac{E(a+\sum_{i=1}^{n}w_{ni}Z_{i})^{-\alpha}}{(a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{-\alpha}}-1=O(\frac{1}{(a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{1-2\beta/r}})$ , where β ≥ 0 $\beta\geq0$ and 2 β / r < 1 $2\beta/r<1$ . Finally, some simulations illustrate the results. We generalize some corresponding results. |
| first_indexed | 2024-04-12T21:31:08Z |
| format | Article |
| id | doaj.art-6d7ff3886f374575a1ad91b7a9e8f96f |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-12T21:31:08Z |
| publishDate | 2016-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-6d7ff3886f374575a1ad91b7a9e8f96f2022-12-22T03:16:02ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-06-012016111110.1186/s13660-016-1099-8The inverse moment for widely orthant dependent random variablesXiaoqin Li0Xi Liu1Wenzhi Yang2Shuhe Hu3School of Mathematical Sciences, Anhui UniversitySchool of Mathematical Sciences, Anhui UniversitySchool of Mathematical Sciences, Anhui UniversitySchool of Mathematical Sciences, Anhui UniversityAbstract In this paper, we investigate approximations of the inverse moment model by widely orthant dependent (WOD) random variables. Let { Z n , n ≥ 1 } $\{Z_{n},n\geq1\}$ be a sequence of nonnegative WOD random variables, and { w n i , 1 ≤ i ≤ n , n ≥ 1 } $\{w_{ni},1\leq i\leq n,n\geq 1\}$ be a triangular array of nonnegative nonrandom weights. If the first moment is finite, then E ( a + ∑ i = 1 n w n i Z i ) − α ∼ ( a + ∑ i = 1 n w n i E Z i ) − α $E(a+ \sum_{i=1}^{n}w_{ni}Z_{i})^{-\alpha}\sim (a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{-\alpha}$ for all constants a > 0 $a>0$ and α > 0 $\alpha>0$ . If the rth moment ( r > 2 $r>2$ ) is finite, then the convergence rate is presented as E ( a + ∑ i = 1 n w n i Z i ) − α ( a + ∑ i = 1 n w n i E Z i ) − α − 1 = O ( 1 ( a + ∑ i = 1 n w n i E Z i ) 1 − 2 β / r ) $\frac{E(a+\sum_{i=1}^{n}w_{ni}Z_{i})^{-\alpha}}{(a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{-\alpha}}-1=O(\frac{1}{(a+\sum_{i=1}^{n}w_{ni}EZ_{i})^{1-2\beta/r}})$ , where β ≥ 0 $\beta\geq0$ and 2 β / r < 1 $2\beta/r<1$ . Finally, some simulations illustrate the results. We generalize some corresponding results.http://link.springer.com/article/10.1186/s13660-016-1099-8inverse moment modelWOD random variablesconvergence ratetriangular array |
| spellingShingle | Xiaoqin Li Xi Liu Wenzhi Yang Shuhe Hu The inverse moment for widely orthant dependent random variables Journal of Inequalities and Applications inverse moment model WOD random variables convergence rate triangular array |
| title | The inverse moment for widely orthant dependent random variables |
| title_full | The inverse moment for widely orthant dependent random variables |
| title_fullStr | The inverse moment for widely orthant dependent random variables |
| title_full_unstemmed | The inverse moment for widely orthant dependent random variables |
| title_short | The inverse moment for widely orthant dependent random variables |
| title_sort | inverse moment for widely orthant dependent random variables |
| topic | inverse moment model WOD random variables convergence rate triangular array |
| url | http://link.springer.com/article/10.1186/s13660-016-1099-8 |
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