1,483 research outputs found

    Combinatorics of free cumulants

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    We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about RR-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals.Comment: 26 pages, Latex2

    Cumulants, free cumulants and half-shuffles

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    Free cumulants were introduced as the proper analog of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, the one of free cumulants is described, and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differ. It is the purpose of the present article to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-)unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical, respectively free, cumulants.Comment: updated and revised version; accepted for publication in PRS

    Cumulants and convolutions via Abel polynomials

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    We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution
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