Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

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 S_n = R⁽ⁿ⁾₁X₁ + ⋅⋅⋅ + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n − 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 − 1/α}/Γ^1/α(α + 1).

KEYWORDS:

• Characteristic function
• limiting distribution
• randomly weighted average
• symmetric α-stable

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F15
• 60E10
• 60E07

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.