The effect of random scale changes on limits of infinitesimal systems
Suppose S={{Xnj, j=1,2,…,kn}} is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple (γ,σ2,M). If {Yj, j=1,2,…} are independent indentically distributed random varia...
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| Format: | Article |
| Language: | English |
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Wiley
1978-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171278000368 |
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| _version_ | 1832555491137421312 |
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| author | Patrick L. Brockett |
| author_facet | Patrick L. Brockett |
| author_sort | Patrick L. Brockett |
| collection | DOAJ |
| description | Suppose S={{Xnj, j=1,2,…,kn}} is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple (γ,σ2,M). If {Yj, j=1,2,…} are independent indentically distributed random variables independent of S, then the system S′={{YjXnj,j=1,2,…,kn}} is obtained by randomizing the scale parameters in S according to the distribution of Y1. We give sufficient conditions on the distribution of Y in terms of an index of convergence of S, to insure that centered sums from S′ be convergent. If such sums converge to a distribution determined by (γ′,(σ′)2,Λ), then the exact relationship between (γ,σ2,M) and (γ′,(σ′)2,Λ) is established. Also investigated is when limit distributions from S and S′ are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law. |
| format | Article |
| id | doaj-art-bfb7584bd2e543ab9ec6e93b4922b000 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1978-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bfb7584bd2e543ab9ec6e93b4922b0002025-02-03T05:48:04ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251978-01-011333937210.1155/S0161171278000368The effect of random scale changes on limits of infinitesimal systemsPatrick L. Brockett0Department of Mathematics, The University of Texas, Austin 78712, Texas, USASuppose S={{Xnj, j=1,2,…,kn}} is an infinitesimal system of random variables whose centered sums converge in law to a (necessarily infinitely divisible) distribution with Levy representation determined by the triple (γ,σ2,M). If {Yj, j=1,2,…} are independent indentically distributed random variables independent of S, then the system S′={{YjXnj,j=1,2,…,kn}} is obtained by randomizing the scale parameters in S according to the distribution of Y1. We give sufficient conditions on the distribution of Y in terms of an index of convergence of S, to insure that centered sums from S′ be convergent. If such sums converge to a distribution determined by (γ′,(σ′)2,Λ), then the exact relationship between (γ,σ2,M) and (γ′,(σ′)2,Λ) is established. Also investigated is when limit distributions from S and S′ are of the same type, and conditions insuring products of random variables belong to the domain of attraction of a stable law.http://dx.doi.org/10.1155/S0161171278000368general central limit theoremproducts of random variables in the domain of attraction of stable lawsLévy spectral function. |
| spellingShingle | Patrick L. Brockett The effect of random scale changes on limits of infinitesimal systems International Journal of Mathematics and Mathematical Sciences general central limit theorem products of random variables in the domain of attraction of stable laws Lévy spectral function. |
| title | The effect of random scale changes on limits of infinitesimal
systems |
| title_full | The effect of random scale changes on limits of infinitesimal
systems |
| title_fullStr | The effect of random scale changes on limits of infinitesimal
systems |
| title_full_unstemmed | The effect of random scale changes on limits of infinitesimal
systems |
| title_short | The effect of random scale changes on limits of infinitesimal
systems |
| title_sort | effect of random scale changes on limits of infinitesimal systems |
| topic | general central limit theorem products of random variables in the domain of attraction of stable laws Lévy spectral function. |
| url | http://dx.doi.org/10.1155/S0161171278000368 |
| work_keys_str_mv | AT patricklbrockett theeffectofrandomscalechangesonlimitsofinfinitesimalsystems AT patricklbrockett effectofrandomscalechangesonlimitsofinfinitesimalsystems |