An inequality for the distance between densities of free convolutions

• Autores: V. Kargin
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 5, 2013, págs. 3241-3260
• Idioma: inglés
• DOI: 10.1214/12-AOP756
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• Resumen

  • This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures μi and νi, i=1,2, are close to each other in terms of the Lé vy metric and if the free convolution μ1⊞μ2 is sufficiently smooth, then ν1⊞ν2 is absolutely continuous, and the densities of measures ν1⊞ν2 and μ1⊞μ2 are close to each other. In particular, convergence in distribution μ(n)1→μ1, μ(n)2→μ2 implies that the density of μ(n)1⊞μ(n)2 is defined for all sufficiently large n and converges to the density of μ1⊞μ2. Some applications are provided, including: (i) a new proof of the local version of the free central limit theorem, and (ii) new local limit theorems for sums of free projections, for sums of ⊞-stable random variables and for eigenvalues of a sum of two N-by-N random matrices.