Explicit free multiplicative law of large numbers
Raouf Fakhfakh
Abstract
Suppose Vν is the pseudo-variance function of the Cauchy–Stieltjes Kernel family K−(ν) generated by a non degenerate probability measure ν on the positive real line. We determine explicitly the law of large numbers μ for free multiplicative convolution given by Haagerup and Möller in terms of the pseudo-variance function Vν. Some examples of calculations of μ are given for probability measures ν of importance in free probability.
Topics & Concepts
Multiplicative functionMathematicsLaw of large numbersProbability measureCauchy distributionMeasure (data warehouse)Free probabilityConvolution (computer science)Variance (accounting)Kernel (algebra)Function (biology)Degenerate energy levelsPure mathematicsRandom variableMathematical analysisStatisticsPhysicsComputer scienceEvolutionary biologyMachine learningAccountingBusinessBiologyArtificial neural networkDatabaseQuantum mechanicsRandom Matrices and ApplicationsSpectral Theory in Mathematical PhysicsBayesian Methods and Mixture Models