9,835 research outputs found
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
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
On a symbolic representation of non-central Wishart random matrices with applications
By using a symbolic method, known in the literature as the classical umbral
calculus, the trace of a non-central Wishart random matrix is represented as
the convolution of the trace of its central component and of a formal variable
involving traces of its non-centrality matrix. Thanks to this representation,
the moments of this random matrix are proved to be a Sheffer polynomial
sequence, allowing us to recover several properties. The multivariate symbolic
method generalizes the employment of Sheffer representation and a closed form
formula for computing joint moments and cumulants (also normalized) is given.
By using this closed form formula and a combinatorial device, known in the
literature as necklace, an efficient algorithm for their computations is set
up. Applications are given to the computation of permanents as well as to the
characterization of inherited estimators of cumulants, which turn useful in
dealing with minors of non-central Wishart random matrices. An asymptotic
approximation of generalized moments involving free probability is proposed.Comment: Journal of Multivariate Analysis (2014
On photon statistics parametrized by a non-central Wishart random matrix
In order to tackle parameter estimation of photocounting distributions,
polykays of acting intensities are proposed as a new tool for computing photon
statistics. As unbiased estimators of cumulants, polykays are computationally
feasible thanks to a symbolic method recently developed in dealing with
sequences of moments. This method includes the so-called method of moments for
random matrices and results to be particularly suited to deal with convolutions
or random summations of random vectors. The overall photocounting effect on a
deterministic number of pixels is introduced. A random number of pixels is also
considered. The role played by spectral statistics of random matrices is
highlighted in approximating the overall photocounting distribution when acting
intensities are modeled by a non-central Wishart random matrix. Generalized
complete Bell polynomials are used in order to compute joint moments and joint
cumulants of multivariate photocounters. Multivariate polykays can be
successfully employed in order to approximate the multivariate Mendel-Poisson
transform. Open problems are addressed at the end of the paper.Comment: 18 pages, in press in Journal of Statistical Planning and Inference,
201
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 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 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
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
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