By Fumio Hiai and Denes Petz
The booklet treats loose chance idea, which has been widely built because the early Eighties. The emphasis is wear entropy and the random matrix version strategy. the amount is a special presentation demonstrating the vast interrelation among the themes. Wigner's theorem and its wide generalizations, equivalent to asymptotic freeness of self reliant matrices, are defined intimately. constant in the course of the booklet is the parallelism among the conventional and semicircle legislation. Voiculescu's multivariate loose entropy conception is gifted with complete proofs and extends the consequences to unitary operators. a few purposes to operator algebras also are given. in keeping with lectures given via the authors in Hungary, Japan, and Italy, the publication is an efficient reference for mathematicians drawn to unfastened chance thought and will function a textual content for a complicated graduate direction.
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Extra info for The semicircle law, free random variables, and entropy
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Yn ), is contained in the a-field generated by Y1 , ... , Yn + 1 , denoted by a(Y1 , ... e. Condition (2) holds since measurability is preserved by addition. Condition (3) follows from Finally, condition (4) holds since Results from probability 21 E[Xn +t l Yo , ... , Yn ] Xn + E (Yn + t ) = Xn. 3 is used. Also note that E[ Yn + 1 I Y0, , Yn ] = E(Yn + 1 ) follows from the assumption that Y0 , Y1 , are independent random variables. �), n 1 , 2, ... } is a martingale. We claim that {( I Xn I , �), n = I , 2, ...
X,), then the measurability condition is automatically satisfied. Recall that a (X1 , ... , Xn), the a-field generated by X1 , ... , X, , is the smallest a-field making X1 , ... , X,1 measurable. Suppose that Y is a random variable defined on the same space and consider the a-field generated by Y, X1 , ... , Xn and denoted by a(Y, X1 , ... , Xn). We have that ••• n . and X1 , ... , X, continue to be measurable with respect to the new a-field a(Y, X1 , ... , Xn). �:, } to be larger than the minimal ones a(X 1 , ...
It must be pointed out a t the outset that a system o ffinite-dimen sional distributions of the form of (7 . I ) does not completely determine the prop erties of the process in the case of an arbitrary index set T. However, the fust step in the general theory of stochastic processes is to construct processes for given fmite-dimensional distributions. I) as a fmite-dimen sional system. I) necessarily satisfies two consistency properties. The first property is the condition ofsymmetry. Let p be a permutation of (I , 2, ...