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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1988-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171288000432 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |