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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1978-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171278000368 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |