ABSTRACT
In this paper we consider a sequence of independent continuous symmetric random variables X1, X2, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages Sn = R(n)1X1 + ⋅⋅⋅ + R(n)nXn, where the random weights R(n)1, …, Rn(n) which are independent of X1, X2, …, Xn, are the cuts of (0, 1) by the n − 1 order statistics from a uniform distribution. Indeed we prove that cnSn converges in distribution to a symmetric α-stable random variable with cn = n1 − 1/α/Γ1/α(α + 1).
Acknowledgments
The author would like to thank the editor-in-chief, the administrator, and anonymous referees for careful reading and for their important and valuable comments and suggestions which greatly improved the paper. He would also like to thank the Yazd University for supporting this research. The author lovingly dedicates this paper to his wife, Forough, who supported him in each step of the way.