A generalization of the global limit theorems of R. P. Agnew
For distribution functions {Fn,n≥0}, the relationship between the weak convergence of Fn to F0 and the convergence of ∫Rϕ(|Fn−F0|)dx to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby general...
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| Format: | Article |
| Language: | English |
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Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171288000432 |
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| _version_ | 1850167554399535104 |
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| author | Andrew Rosalsky |
| author_facet | Andrew Rosalsky |
| author_sort | Andrew Rosalsky |
| collection | DOAJ |
| description | For distribution functions {Fn,n≥0}, the relationship between the weak convergence of Fn to F0 and the convergence of ∫Rϕ(|Fn−F0|)dx to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew wherein ϕ(t)=|t|r. The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line. |
| format | Article |
| id | doaj-art-e7017bebf88c4a82bd846f750a26a380 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1988-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e7017bebf88c4a82bd846f750a26a3802025-08-20T02:21:10ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251988-01-0111236537410.1155/S0161171288000432A generalization of the global limit theorems of R. P. AgnewAndrew Rosalsky0Department of Statistics, University of Florida, Gainesville 32611, Florida, USAFor distribution functions {Fn,n≥0}, the relationship between the weak convergence of Fn to F0 and the convergence of ∫Rϕ(|Fn−F0|)dx to 0 is studied where ϕ is a nonnegative, nondecreasing function. Sufficient and, separately, necessary conditions are given for the latter convergence thereby generalizing the so-called global limit theorems of Agnew wherein ϕ(t)=|t|r. The sufficiency results are shown to be sharp and, as a special case, yield a global version of the central limit theorem for independent random variables obeying the Liapounov condition. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to Lebesgue measure on the line.http://dx.doi.org/10.1155/S0161171288000432distribution functionglobal limit theoremweak convergencecomplete convergencealmost everywhere convergenceuniform convergencesums of independent random variablescentral limit theoremLiapounov condition. |
| spellingShingle | Andrew Rosalsky A generalization of the global limit theorems of R. P. Agnew International Journal of Mathematics and Mathematical Sciences distribution function global limit theorem weak convergence complete convergence almost everywhere convergence uniform convergence sums of independent random variables central limit theorem Liapounov condition. |
| title | A generalization of the global limit theorems of R. P. Agnew |
| title_full | A generalization of the global limit theorems of R. P. Agnew |
| title_fullStr | A generalization of the global limit theorems of R. P. Agnew |
| title_full_unstemmed | A generalization of the global limit theorems of R. P. Agnew |
| title_short | A generalization of the global limit theorems of R. P. Agnew |
| title_sort | generalization of the global limit theorems of r p agnew |
| topic | distribution function global limit theorem weak convergence complete convergence almost everywhere convergence uniform convergence sums of independent random variables central limit theorem Liapounov condition. |
| url | http://dx.doi.org/10.1155/S0161171288000432 |
| work_keys_str_mv | AT andrewrosalsky ageneralizationofthegloballimittheoremsofrpagnew AT andrewrosalsky generalizationofthegloballimittheoremsofrpagnew |