The semicircle law, free random variables and entropy by Fumio Hiai and Denes Petz

By Fumio Hiai and Denes Petz

The booklet treats loose likelihood concept, which has been generally constructed because the early Nineteen Eighties. The emphasis is wear entropy and the random matrix version strategy. the quantity is a different presentation demonstrating the large interrelation among the subjects. Wigner's theorem and its huge generalizations, similar to asymptotic freeness of self reliant matrices, are defined intimately. constant through the booklet is the parallelism among the traditional and semicircle legislation. Voiculescu's multivariate loose entropy thought 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 e-book is an efficient reference for mathematicians attracted to loose likelihood idea and will function a textual content for a sophisticated graduate direction

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5) as required. 6(2) suggests that we generalise the definition of infinite divisibility as follows: µ ∈ M1 (Rd ) is infinitely divisible if it has a convolution nth root in M1 (Rd ) for each n ∈ N. 7 Show that µ ∈ M1 (Rd ) is infinitely divisible if and only if for each n ∈ N there exists µ1/n ∈ M1 (Rd ) for which φµ (x) = φµ1/n (x) n for each x ∈ Rd . 7 is unique when µ is infinitely divisible. Moreover, in this case the complexvalued function φµ always has a ‘distinguished’ nth root, which we denote 1/n by φµ ; this is the characteristic function of µ1/n (see Sato [274], pp.

J=1 In particular, a sequence of random variables (X n , n ∈ N) is said to be independent if (σ (X n ), n ∈ N) is independent in the above sense. d. e. the laws ( p X n , n ∈ N) are identical probability measures. We say that a random variable X and a sub-σ -algebra G of F are independent if σ (X ) and G are independent. s. Now let {(S1 , F1 , µ1 ), . . , (Sn , Fn , µn )} be a family of measure spaces. We define their product to be the space (S, F, µ), where S is the Cartesian product S1 × S2 × · · · × Sn , F = F1 ⊗ F2 ⊗ · · · ⊗ Fn is the smallest σ -algebra containing all sets of the form A1 × A2 × · · · × An for which each Ai ∈ Fi and µ = µ1 × µ2 × · · · × µn is the product measure for which n µ(A1 × A2 × · · · × An ) = µ(Ai ).

2 Definition of infinite divisibility Let X be a random variable taking values in Rd with law µ X . d. random variables (n) (n) Y1 , . . , Yn such that X = Y1(n) + · · · + Yn(n) . 5) Let φ X (u) = E(ei(u,X ) ) denote the characteristic function of X , where u ∈ Rd . More generally, if µ ∈ M1 (Rd ) then φµ (u) = Rd ei(u,y) µ(dy). 6 The following are equivalent: (1) X is infinitely divisible; (2) µ X has a convolution nth root that is itself the law of a random variable, for each n ∈ N; (3) φ X has an nth root that is itself the characteristic function of a random variable, for each n ∈ N.

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