Addition of freely independent random variables

Abstract

A direct proof is given of Voiculescu's addition theorem for freely independent real-valued random variables, using resolvents of self-adjoint operators. In contrast to the original proof, no assumption is made on the existence of moments above the second. g: 1992 Academic Press. Inc The concept of independent random variables lies at the heart of classical probability. Via independent sequences it leads to the Gauss and Poisson distributions, and via independent increments of a process to stochastic calculus. Classical, commutative independence of random variables amounts to a factorisation property of probability spaces. Algebraically this corresponds to a tensor product decomposition of function algebras. At the opposite, non-commutative extreme Voicuiescu discovered in 1983 the notion of "free independence" of random variables, which corresponds to a free product of von Neumann algebras [Vail]. He showed that this notion leads naturally to analogues of the Gauss and Poisson distributions, very different in form from the classical ones [Voil, Voi5]. For instance the free analogue of the Gauss curve is a semi-ellipse. He also showed that n x n-matrices whose entries are independent Gaussian random variables become freely independent in the limit of large n [Voi4]. This explains the older result [Wigl, Wig2, Arnl, Arn2, Wac, Jon] that the eigenvalue distributions of such random matrices with probability 1 tend to the semi-ellipse form as n + co. The relevance of free independence for non-commutative probability theory was realised by Kiimmerer, who, with Speicher, developed a stochastic calculus based on free independence [KSp, Spel, SpeZ]. In this paper we consider the addition problem: What is the probability distribution p of the sum X, + X1 of two freely independent random variables, given the distributions p, and I(~ of the summands? This problem 409

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