Random arithmetic-geometric means and random pi: observations and conjectures

Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Limits Laws for Geometric Means of Free Random Variables

Let {Tk}k=1 be a family of ∗–free identically distributed operators in a finite von Neumann algebra. In this work we prove a multiplicative version of the free central limit Theorem. More precisely, let Bn = T ∗ 1 T ∗ 2 . . . T ∗ nTn . . . T2T1 then Bn is a positive operator and B 1/2n n converges in distribution to an operator Λ. We completely determine the probability distribution ν of Λ from...

Analysis of Random Processes with Fuzzy Observations Using Discretization Method

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Limits Laws for Geometric Means of Free Positive Random Variables

Let {ak}k=1 be free identically distributed positive non–commuting random variables with probability measure distribution μ. In this paper we proved a multiplicative version of the Free Central Limit Theorem. More precisely, let bn = a 1/2 1 a 1/2 2 . . . an . . . a 1/2 2 a 1/2 1 then bn is a positive operator with the same moments as xn = a1a2 . . . an and b 1/2n n converges in distribution to...

Mixed Arithmetic and Geometric Means and Related Inequalities

Mixed arithmetic and geometric means, with and without weights, are both considered. Related to mixed arithmetic and geometric means, the following three types of inequalities and their generalizations, from three variables to a general n variables, are studied. For arbitrary x, y, z ≥ 0 we have [ x + y + z 3 (xyz) ]1/2 ≤ ( x + y 2 · y + z 2 · z + x 2 )1/3 , (A) 1 3 (√ xy + √ yz + √ zx ) ≤ 1 2 ...