180 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
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
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
Symbolic Calculus in Mathematical Statistics: A Review
In the last ten years, the employment of symbolic methods has substantially
extended both the theory and the applications of statistics and probability.
This survey reviews the development of a symbolic technique arising from
classical umbral calculus, as introduced by Rota and Taylor in The
usefulness of this symbolic technique is twofold. The first is to show how new
algebraic identities drive in discovering insights among topics apparently very
far from each other and related to probability and statistics. One of the main
tools is a formal generalization of the convolution of identical probability
distributions, which allows us to employ compound Poisson random variables in
various topics that are only somewhat interrelated. Having got a different and
deeper viewpoint, the second goal is to show how to set up algorithmic
processes performing efficiently algebraic calculations. In particular, the
challenge of finding these symbolic procedures should lead to a new method, and
it poses new problems involving both computational and conceptual issues.
Evidence of efficiency in applying this symbolic method will be shown within
statistical inference, parameter estimation, L\'evy processes, and, more
generally, problems involving multivariate functions. The symbolic
representation of Sheffer polynomial sequences allows us to carry out a
unifying theory of classical, Boolean and free cumulants. Recent connections
within random matrices have extended the applications of the symbolic method.Comment: 72 page
Probability measures, L\'{e}vy measures and analyticity in time
We investigate the relation of the semigroup probability density of an
infinite activity L\'{e}vy process to the corresponding L\'{e}vy density. For
subordinators, we provide three methods to compute the former from the latter.
The first method is based on approximating compound Poisson distributions, the
second method uses convolution integrals of the upper tail integral of the
L\'{e}vy measure and the third method uses the analytic continuation of the
L\'{e}vy density to a complex cone and contour integration. As a by-product, we
investigate the smoothness of the semigroup density in time. Several concrete
examples illustrate the three methods and our results.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6114 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Parametric estimation of the driving L\'evy process of multivariate CARMA processes from discrete observations
We consider the parametric estimation of the driving L\'evy process of a
multivariate continuous-time autoregressive moving average (MCARMA) process,
which is observed on the discrete time grid . Beginning with a
new state space representation, we develop a method to recover the driving
L\'evy process exactly from a continuous record of the observed MCARMA process.
We use tools from numerical analysis and the theory of infinitely divisible
distributions to extend this result to allow for the approximate recovery of
unit increments of the driving L\'evy process from discrete-time observations
of the MCARMA process. We show that, if the sampling interval is chosen
dependent on , the length of the observation horizon, such that
converges to zero as tends to infinity, then any suitable generalized
method of moments estimator based on this reconstructed sample of unit
increments has the same asymptotic distribution as the one based on the true
increments, and is, in particular, asymptotically normally distributed.Comment: 38 pages, four figures; to appear in Journal of Multivariate Analysi
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