Abstract
We show that there is a faster convergence in the free central limit theorem for measures of bounded support when we have vanishing free cumulants. We give estimates for the extremes of the support and the density of the converging measure. As a consequence, we obtain a more precise Berry–Esseen type estimate than previous results.
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References
Arizmendi, O., Vargas, C.: Norm convergence in non-commutative central limit theorems: combinatorial approach. Contemp. Math. (2016). https://doi.org/10.1090/conm/657/13089
Bercovici, H., Voiculescu, D.: Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Relat. Fields (1995). https://doi.org/10.1007/BF01204215
Chistyakov, G.P., Götze, F.: Asymptotic expansions in the CLT in free probability. Probab. Theory Relat. Fields (2013). https://doi.org/10.1007/s00440-012-0451-2
Chistyakov, G.P., Götze, F.: Limit theorems in free probability theory. I. Ann. Probab. (2008). https://doi.org/10.1214/009117907000000051
Huang, H.W.: Supports of measures in a free additive convolution semigroup. Int. Math. Res. Not. (2015). https://doi.org/10.1093/imrn/rnu064
Klartag, B., Sodin, S.: Variations on the Berry–Esseen Theorem. Theory Probab. Appl. (2012). https://doi.org/10.1137/S0040585X97985522
Mingo, J.A., Speicher, R.: Free Probability and Random Matrices. Springer, New York (2017)
Wang, J.C.: Local limit theorems in free probability theory. Ann. Probab. (2010). https://doi.org/10.1214/09-AOP505
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The author appreciates the support of CONACYT through the program “Estancias posdoctorales por México”. The author also express sincere thanks to Octavio Arizmendi for useful discussions.
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Communicated by Ilwoo Cho.
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Salazar, M. Faster Convergence in the Free Central Limit Theorem. Complex Anal. Oper. Theory 17, 66 (2023). https://doi.org/10.1007/s11785-023-01373-5
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DOI: https://doi.org/10.1007/s11785-023-01373-5