6,831 research outputs found
Polynomial traces and elementary symmetric functions in the latent roots of a non-central Wishart matrix
Hypergeometric functions and zonal polynomials are the tools usually
addressed in the literature to deal with the expected value of the elementary
symmetric functions in non-central Wishart latent roots. The method here
proposed recovers the expected value of these symmetric functions by using the
umbral operator applied to the trace of suitable polynomial matrices and their
cumulants. The employment of a suitable linear operator in place of
hypergeometric functions and zonal polynomials was conjectured by de Waal in
1972. Here we show how the umbral operator accomplishes this task and
consequently represents an alternative tool to deal with these symmetric
functions. When special formal variables are plugged in the variables, the
evaluation through the umbral operator deletes all the monomials in the latent
roots except those contributing in the elementary symmetric functions.
Cumulants further simplify the computations taking advantage of the convolution
structure of the polynomial trace. Open problems are addressed at the end of
the paper
Multivariate Bernoulli and Euler polynomials via L\'evy processes
By a symbolic method, we introduce multivariate Bernoulli and Euler
polynomials as powers of polynomials whose coefficients involve multivariate
L\'evy processes. Many properties of these polynomials are stated
straightforwardly thanks to this representation, which could be easily
implemented in any symbolic manipulation system. A very simple relation between
these two families of multivariate polynomials is provided
On multivariable cumulant polynomial sequences with applications
A new family of polynomials, called cumulant polynomial sequence, and its
extensions to the multivariate case is introduced relied on a purely symbolic
combinatorial method. The coefficients of these polynomials are cumulants, but
depending on what is plugged in the indeterminates, either sequences of moments
either sequences of cumulants can be recovered. The main tool is a formal
generalization of random sums, also with a multivariate random index and not
necessarily integer-valued. Applications are given within parameter
estimations, L\'evy processes and random matrices and, more generally, problems
involving multivariate functions. The connection between exponential models and
multivariable Sheffer polynomial sequences offers a different viewpoint in
characterizing these models. Some open problems end the paper.Comment: 17 pages, In pres
On a representation of time space-harmonic polynomials via symbolic L\'evy processes
In this paper, we review the theory of time space-harmonic polynomials
developed by using a symbolic device known in the literature as the classical
umbral calculus. The advantage of this symbolic tool is twofold. First a moment
representation is allowed for a wide class of polynomial stochastic involving
the L\'evy processes in respect to which they are martingales. This
representation includes some well-known examples such as Hermite polynomials in
connection with Brownian motion. As a consequence, characterizations of many
other families of polynomials having the time space-harmonic property can be
recovered via the symbolic moment representation. New relations with
Kailath-Segall polynomials are stated. Secondly the generalization to the
multivariable framework is straightforward. Connections with cumulants and Bell
polynomials are highlighted both in the univariate case and in the multivariate
one. Open problems are addressed at the end of the paper
On some applications of a symbolic representation of non-centered L\'evy processes
By using a symbolic technique known in the literature as the classical umbral
calculus, we characterize two classes of polynomials related to L\'evy
processes: the Kailath-Segall and the time-space harmonic polynomials. We
provide the Kailath-Segall formula in terms of cumulants and we recover simple
closed-forms for several families of polynomials with respect to not centered
L\'evy processes, such as the Hermite polynomials with the Brownian motion, the
Poisson-Charlier polynomials with the Poisson processes, the actuarial
polynomials with the Gamma processes, the first kind Meixner polynomials with
the Pascal processes, the Bernoulli, Euler and Krawtchuk polynomials with
suitable random walks
On the computation of classical, boolean and free cumulants
This paper introduces a simple and computationally efficient algorithm for
conversion formulae between moments and cumulants. The algorithm provides just
one formula for classical, boolean and free cumulants. This is realized by
using a suitable polynomial representation of Abel polynomials. The algorithm
relies on the classical umbral calculus, a symbolic language introduced by Rota
and Taylor in 1994, that is particularly suited to be implemented by using
software for symbolic computations. Here we give a MAPLE procedure. Comparisons
with existing procedures, especially for conversions between moments and free
cumulants, as well as examples of applications to some well-known distributions
(classical and free) end the paper.Comment: 14 pages. in press, Applied Mathematics and Computatio
A symbolic method for k-statistics
Trough the classical umbral calculus, we provide new, compact and easy to
handle expressions of k-statistics, and more in general of U-statistics. In
addition such a symbolic method can be naturally extended to multivariate case
and to generalized k-statistics.Comment: Extended abstract with corrected typos and change conten
Multivariate time-space harmonic polynomials: a symbolic approach
By means of a symbolic method, in this paper we introduce a new family of
multivariate polynomials such that multivariate L\'evy processes can be dealt
with as they were martingales. In the univariate case, this family of
polynomials is known as time-space harmonic polynomials. Then, simple
closed-form expressions of some multivariate classical families of polynomials
are given. The main advantage of this symbolic representation is the plainness
of the setting which reduces to few fundamental statements but also of its
implementation in any symbolic software. The role played by cumulants is
emphasized within the generalized Hermite polynomials. The new class of
multivariate L\'evy-Sheffer systems is introduced.Comment: In pres
CUB models: a preliminary fuzzy approach to heterogeneity
In line with the increasing attention paid to deal with uncertainty in
ordinal data models, we propose to combine Fuzzy models with \cub models within
questionnaire analysis. In particular, the focus will be on \cub models'
uncertainty parameter and its interpretation as a preliminary measure of
heterogeneity, by introducing membership, non-membership and uncertainty
functions in the more general framework of Intuitionistic Fuzzy Sets. Our
proposal is discussed on the basis of the Evaluation of Orientation Services
survey collected at University of Naples Federico II.Comment: 10 pages, invited contribution at SIS2016 (Salerno, Italy), in
SIS2016 proceeding
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