Central Limit for the Product of Free Random Variables

Keang-Po Ho

Central Limit for the Product of Free Random Variables

Operator Algebras 2014-10-13 v3 Probability

Authors: Keang-Po Ho

Abstract

The central limit for the product of free random variables are studied by evaluating all the moments of the limit distribution. The logarithm of the central limit is found to be the same as the sum of two independent free random variables: one semicircularly distributed and another uniformly distributed. The logarithm of central limit has a moment-generating function of $\exp(\xi^2 s/2) {_{1}F_{1}}\left(1-s; 2; -\xi^2 s \right)$ .

Keywords

free random variables central limit theorem product of random variables semicircular distribution moment-generating function

Cite

@article{arxiv.1101.5220,
  title  = {Central Limit for the Product of Free Random Variables},
  author = {Keang-Po Ho},
  journal= {arXiv preprint arXiv:1101.5220},
  year   = {2014}
}

Comments

correct some typos, 13 pages, 1 figure

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