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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 120 (7 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Stable Laws and Domains of Attraction in Free Probability Theory
, 1999
"... In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our wor ..."
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Cited by 87 (1 self)
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In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices
, 2011
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On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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Cited by 36 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
Free Diffusions, Free Entropy And Free Fisher Information
"... . Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free ..."
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Cited by 32 (1 self)
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. Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t = dS t \Gamma 1 2 f(X t )dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t ! 1, towards the master field (i.e. the Gibbs state). 1. Introduction The purpose of this paper is to start the study of diffusion equations where the driving noise is a free brownian motion. Reasons for considering such equations will be explained in the next sections of this introduction. 1.1 Gibbs states and diffusion theory. Let V be a C 2 function on R d , with Z = Z R d e \GammaV (x) dx ! 1: The probability measur...
Commutators of Free Random Variables
 Duke Math. J
, 1998
"... Let A be a unital C algebra, given together with a specied state ' : A ! C. ..."
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Cited by 32 (5 self)
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Let A be a unital C algebra, given together with a specied state ' : A ! C.
A free probability analogue of the Wasserstein metric on the tracestate space
 Geom. Funct. Anal
"... Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0 ..."
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Cited by 28 (0 self)
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Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0
Free analysis questions. I. Duality transform for the coalgebra of ∂X
 B, Int. Math. Res. Not
"... Preliminary Version A duality transform for the coalgebra of the free difference quotient derivationmultiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalizatio ..."
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Cited by 27 (1 self)
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Preliminary Version A duality transform for the coalgebra of the free difference quotient derivationmultiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation. 1
Limit theorems in free probability theory. I
 I ARXIV:MATH. OA/0602219 V
, 2006
"... Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for nonidentically distributed random variables in classical Probability Theory. ..."
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Cited by 21 (5 self)
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Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for nonidentically distributed random variables in classical Probability Theory.