471 research outputs found
Cumulants in Noncommutative Probability Theory IV. De Finetti's Theorem and -Inequalities
In this paper we collect a few results about exchangeability systems in which
crossing cumulants vanish, which we call noncrossing exchangeability systems.
The main result is a free version of De Finetti's theorem, characterising
amalgamated free products as noncrossing exchangeability systems which satisfy
a so-called weak singleton condition. The main tool in the proof is an
-inequality with uniformly bounded constants for i.i.d. sequences in
noncrossing exchangeability systems.Comment: 33 pages, AMS LaTeX; Brillinger's formula transferred to separate
pape
Cumulants in noncommutative probability II. Generalized Gaussian random variables
We continue the investigation of noncommutative cumulants.
In this paper various characterizations of noncommutative Gaussian random
variables are proved.Comment: 14 pages, AMS-LaTeX; many corrections according to the referee; to
appear in Prob Th Rel Field
Cumulants, lattice paths, and orthogonal polynomials
A formula expressing free cumulants in terms of the Jacobi parameters of the
corresponding orthogonal polynomials is derived. It combines Flajolet's theory
of continued fractions and Lagrange inversion. For the converse we discuss
Gessel-Viennot theory to express Hankel determinants in terms of various
cumulants.Comment: 11 pages, AMS LaTeX, uses pstricks; revised according to referee's
suggestions, in particular cut down last section and corrected some wrong
attribution
On the computation of spectra in free probability
We use free probability techniques to compute borders of spectra of non
hermitian operators in finite von Neumann algebras which arise as `free sums'
of `simple' operators. To this end, the resolvent is analyzed with the aid of
the Haagerup inequality. Concrete examples coming from reduced C*-algebras of
free product groups and leading to systems of polynomial equations illustrate
the approach.Comment: LaTeX2e, 15 pages, 3 figure
On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs
We show that the Plancherel measure of the lamplighter random walk on a graph
coincides with the expected spectral measure of the absorbing random walk on
the Bernoulli percolation clusters. In the subcritical regime the spectrum is
pure point and we construct a complete orthonormal basis of finitely supported
eigenfunctions.Comment: notational improvements and minor corrections, 6 pages; complement to
arXiv:0712.313
Free Lamplighter Groups and a Question of Atiyah
We compute the von Neumann dimensions of the kernels of adjacency operators
on free lamplighter groups and show that they are irrational, thus providing an
elementary constructive answer to a question of Atiyah.Comment: AMSLaTeX, 10 pages, to appear in Amer J Mat
Cumulants, Spreadability and the Campbell-Baker-Hausdorff Series
We define spreadability systems as a generalization of exchangeability
systems in order to unify various notions of independence and cumulants known
in noncommutative probability. In particular, our theory covers monotone
independence and monotone cumulants which do not satisfy exchangeability. To
this end we study generalized zeta and M\"obius functions in the context of the
incidence algebra of the semilattice of ordered set partitions and prove an
appropriate variant of Faa di Bruno's theorem. With the aid of this machinery
we show that our cumulants cover most of the previously known cumulants. Due to
noncommutativity of independence the behaviour of these cumulants with respect
to independent random variables is more complicated than in the exchangeable
case and the appearance of Goldberg coefficients exhibits the role of the
Campbell-Baker-Hausdorff series in this context. In a final section we exhibit
an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants
in a particular spreadability system, thus providing a new derivation of the
Goldberg coefficients.Comment: some minor corrections, 48 page
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