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1
Content available remote A constant regression characterization of the Marchenko-Pastur law
Szpojankowski K.
Probability and Mathematical Statistics
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2016
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Vol. 36, Fasc. 1
137--145
EN
In this paper, Lukacs type characterization of Marchenko-Pastur distribution in free probability is studied. We prove that for free X and Y, if conditional moments of order 1 and −1 of (X + Y)−1/2 X(X + Y)−1/2 given X + Y are constant, then X and Y follow the Marchenko-Pastur distribution.
2
Content available Free probability on Hecke algebras and certain group C*-algebras induced by Hecke algebras
Cho I.
Opuscula Mathematica
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2016
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Vol. 36, no. 2
153--187
EN
In this paper, by establishing free-probabilistic models on the Hecke algebras [formula] induced by p-adic number fields Qp, we construct free probability spaces for all primes p. Hilbert-space representations are induced by such free-probabilistic structures. We study C*-algebras induced by certain partial isometries realized under the representations.
3
Content available Certain group dynamical systems induced by Hecke algebras
Cho I.
Opuscula Mathematica
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2016
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Vol. 36, no. 3
337--373
EN
In this paper, we study dynamical systems induced by a certain group [formula] embedded in the Hecke algebra H(Gp) induced by the generalized linear group Gp = GL2(Qp) over the p-adic number fields Qp for a fixed prime p. We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms ol free probability on the Hecke algebra H(Gp).
4
Content available Free probability induced by electric resistance networks on energy Hilbert spaces
Cho I., Jorgensen P. E. T.
Opuscula Mathematica
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2011
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Vol. 31, no. 4
549-598
EN
We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space Hε. From Hε, one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on Hε. With the use of our ERN-groupoid, we show that Hε may be derived as a representation Hilbert space of a universal representation of a groupoid algebra [formula], and we display other representations. Among our applications, we identify a free structure of [formula] in terms of the energy form.
5
Content available remote Free exponential families as kernel families
Bryc W.
Demonstratio Mathematica
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2009
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Vol. 42, nr 3
657-672
EN
Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from the approach analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel 1/(1 - Qx) instead of the exponential kernel exp(Qx). We use this approach to re-derive some known results and to study further similarities with exponential families and reproductive exponential models.
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