92 research outputs found
Free Martingale polynomials for stationary Jacobi processes
We generalize a previous result concerning free martingale polynomials for
the stationary free Jacobi process of parameters . Hopelessly, apart from the case , the polynomials we derive
are no longer orthogonal with respect to the spectral measure. As a matter of
fact, we use the multiplicative renormalization to write down the corresponding
orthogonality measure.Comment: page number : 1
First hitting time of the boundary of a wedge of angle by a radial Dunkl process
In this paper, we derive an integral representation for the density of the
reciprocal of the first hitting time of the boundary of a wedge of angle
by a radial Dunkl process with equal multiplicity values. Not only this
representation readily yields the non negativity of the density, but also
provides an analogue of Dufresne's result on the distribution of the first
hitting time of zero by a Bessel process and a generalization of the
Vakeroudis-Yor's identity satisfied by the first exit time from a wedge by a
planar Brownian motion. We also use a result due to Spitzer on the angular part
of the planar Brownian motion to prove a representation of the tail
distribution of its first exit time from a dihedral wedge through the square
wave function.Comment: Title is changed, many corrections, new result
Generalized Bessel function of Type D
We write down the generalized Bessel function associated with the root system
of type by means of multivariate hypergeometric series. Our hint comes from
the particular case of the Brownian motion in the Weyl chamber of type .Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Radial Dunkl Processes : Existence and uniqueness, Hitting time, Beta Processes and Random Matrices
We begin with the study of some properties of the radial Dunkl process
associated to a reduced root system . It is shown that this diffusion is the
unique strong solution for all of a SDE with singular drift. Then,
we study , the first hitting time of the positive Weyl chamber : we prove,
via stochastic calculus, a result already obtained by Chybiryakov on the
finiteness of . The second and new part deals with the law of for
which we compute the tail distribution, as well as some insight via stochastic
calculus on how root systems are connected with eigenvalues of standard
matrix-valued processes. This gives rise to the so-called -processes.
The ultraspherical -Jacobi case still involves a reduced root system
while the general case is closely connected to a non reduced one. This process
lives in a convex bounded domain known as principal Weyl alcove and the strong
uniqueness result remains valid. The last part deals with the first hitting
time of the alcove's boundary and the semi group density which enables us to
answer some open questions.Comment: 33 page
On generalized Cauchy-Stieltjes transforms of some Beta distributions
We express generalized Cauchy-Stieltjes transforms of some particular Beta
distributions (of ultraspherical type generating functions for orthogonal
polynomials) as a powered Cauchy-Stieltjes transform of some measure. For
suitable values of the power parameter, the latter measure turns out to be a
probability measure and its density is written down using Markov transforms.
The discarded values give a negative answer to a deformed free probability
unless a restriction on the power parameter is made. A particular symmetric
distribution interpolating between Wigner and arcsine distributions is
obtained. Its moments are expressed through a terminating hypergeometric series
interpolating between Catalan and shifed Catalan numbers. for small values of
the power parameter, the free cumulants are computed. Interesting opne problems
related to a deformed representation theory of the infinite symmetric group and
to a deformed Bozejko's convolution are discussed
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