Algebras with two multiplications and their cumulants

@article{Burchardt2019AlgebrasWT,
  title={Algebras with two multiplications and their cumulants},
  author={Adam Burchardt},
  journal={Journal of Algebraic Combinatorics},
  year={2019}
}
  • Adam Burchardt
  • Published 25 March 2018
  • Mathematics
  • Journal of Algebraic Combinatorics
Cumulants are a notion that comes from the classical probability theory; they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting… 
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References

SHOWING 1-10 OF 43 REFERENCES
Free nested cumulants and an analogue of a formula of Brillinger
We prove a free analogue of Brillinger's formula (sometimes called "law of total cumulance") which expresses classical cumulants in terms of conditioned cumulants. As expected, the formula is…
A combinatorial formula for Macdonald cumulants
Macdonald cumulants are symmetric functions that generalize Macdonald polynomials. We prove a combinatorial formula for them which extends the celebrated formula of Haglund for Macdonald polynomials.…
Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems
Abstract.Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is…
Combinatorics of Free Cumulants
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…
Free cumulants, Schröder trees, and operads
GAUSSIAN FLUCTUATIONS OF YOUNG DIAGRAMS AND STRUCTURE CONSTANTS OF JACK CHARACTERS
In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first- and second-order asymptotics of the bulk of a random Young…
Jack polynomials and orientability generating series of maps
We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who…
Second order cumulants of products
We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of…
Cumulants of Jack symmetric functions and b-conjecture (extended abstract)
International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous…
CUMULANTS AND PARTITION LATTICES1
Summary The (joint) cumulant of a set of (possibly coincident) random variables is defined as an alternating sum of moments with appropriate integral coefficients. By exploiting properties of the…
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