Free Random Variables

By O. Johnson. 2004

948 citations recorded by the source corpus.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3746f8309bcdecb8073ab5206602b5a70a4a8c28

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In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Group measure space decomposition of II1 factors and W*-superrigidity (2009, background): “Denote by M = M1 ∗P M2 the amalgamated free product with respect to the unique trace preserving conditional expectations (see [Po93] and [VDN92]) and still denote by τ the canonical”
Non injectivity of the q-deformed von Neumann algebra (2004, background): “ ⊗ ξ. Recall that the free von Neumann algebra Γ 0(HR) (corresponding to T = 0) is not injective as soon as n = dimHR ≥ 2, for Γ 0(HR) is isomorphic to the free group von Neumann algebra VN(F n) (cf. [18]). The main result of this paper solves the above problem. To explain the idea of our proof we ﬁ”
Testing whether linear equations are causal: A free probability theory approach (2011, methodology): “To prove Theorem 1 we use the following results which are based on free probability theory (Voiculescu et al., 1992; Voiculescu, 1997).”
Generating random density matrices (2010, neutral): “1 ([41] and [42]) Let U1,N , .”
Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability (2002, result): “··· ,ad k)d is asymptotically free as d→ ∞ iﬀ it converges in law towards a free random variable. By now we are able to state the main result of this section. Note that it was originally contained in [VDN92] under stronger hypotheses. Feng Xu [Xu97] obtains an analogous result using geometric method”
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution (2007, methodology): “They are defined in order to reflect the addition and respectively the multiplication of free random variables; we refer to Chapter 3 of [23] or Chapters 2, 3 in [22] for a precise description of how this goes.”
Rosenthal Type inequalities for free chaos. (2005, background): “The adjoint map l(e) is called the annihilation operator on F(H), see [45] for more details.”
Gaussian fluctuations of characters of symmetric groups and of Young diagrams (2005, methodology): “his way we prove a generalization of Kerov’s central limit theorem for a very wide class of representations. 1.2. Genus expansion, random matrices and free probability. Free probability of Voiculescu [VDN92] is a non-commutative probability theory which turned out to be very successful in describing”
Financial Applications of Random Matrix Theory: Old Laces and New Pieces (2005, positive): “The quantity R(x) = B(x) − 1/x is the ‘R-transform’ of G, and is known to be additive [5, 11, 12].”
Outliers in the spectrum of large deformed unitarily invariant models (2012, methodology): “If both of μ and ν have zero first moment, then μ ν is the Haar (uniform) distribution on T, see [42].”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, positive): “The type classiﬁcation of these factors is the following: (1) M is a type II 1 factor if and only if the representation ( U t ) is trivial: in that case the functor Γ is Voiculescu’s free Gaussian functor [56].”