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Central Limit for the Product of Free Random Variables

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arxiv 1101.5220 v3 pith:Q3FFGPFK submitted 2011-01-27 math.OA math.PR

Keang-Po Ho This is my paper · ORCID
classification math.OAmath.PR
keywords limitcentralfreerandomvariablesdistributedlogarithmproduct
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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 $\exp(\xi^2 s/2) {_{1}F_{1}}\left(1-s; 2; -\xi^2 s \right)$.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Rates of convergence in the Free Multiplicative Central Limit Theorem

    math.OA 2025-05 conditional novelty 8.0 of 10

    First quantitative convergence rates for the free multiplicative CLT in Wasserstein and Kolmogorov distances.

  2. Hua-Pickrell diffusions and differential equations related with pseudo-Jacobi polynomials

    math.PR 2026-02 conditional novelty 7.0 of 10

    For Hua-Pickrell diffusions, the large-N empirical limits are independent of β, and the frozen β=∞ limits are the zeros of pseudo-Jacobi polynomials.

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