THE EFFECT OF RANDOM SCALE CHANGES ON LIMITS OF INFINITESIMAL SYSTEMS

Suppose S {{X_(nj), j=1,2,...,k_n}} 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, σ^2,M). If {Y_j, j=1,2,... } are independent indentically distributed random variables independent of S, then the system S' {{Y_jX_(nj), j=1,2,...,k_n}} is obtained by randomizing the scale parameters in S according to the distribution of Y_1. 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,ʌ), then the exact relationship between (γ, σ^2,M) and (γ', (σ')^2,ʌ) 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.

並列關鍵字

General central limit theorem ； products of random variables in the domain of attraction of stable laws ； Lvy spectral function

國際替代計量

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