108 research outputs found
Wiener ( Ornstein-Uhlenbeck processes. A generalization of known processes
We collect, scattered through literature, as well as we prove some new
properties of two Markov processes that in many ways resemble Wiener and
Ornstein--Uhlenbeck processes. Although processes considered in this paper were
defined either in non-commutative probability context or through quadratic
harnesses we define them once more as so to say 'continuous time '
generalization of a simple, symmetric, discrete time process satisfying simple
conditions imposed on the form of its first two conditional moments. The finite
dimensional distributions of the first one (say X=(X_{t})_{t\geq0} called
q-Wiener) depends on one parameter q\in(-1,1] and of the second one (say
Y=(Y_{t})_{t\inR} called ({\alpha},q)- Ornstein--Uhlenbeck) on two parameters
({\alpha},q)\in(0,\infty)\times(-1,1]. The first one resembles Wiener process
in the sense that for q=1 it is Wiener process but also that for |q|<1 and
\foralln\geq1: t^{n/2}H_{n}(X_{t}/\surdt|q), where (H_{n})_{n\geq0} are the so
called q-Hermite polynomials, are martingales. It does not have however neither
independent increments not allows continuous sample path modification. The
second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU
process. For |q|<1 it is also stationary with correlation function equal to
exp(-{\alpha}|t-s|) and has many properties resembling those of its classical
version. We think that these process are fascinating objects to study posing
many interesting, open questions.Comment: 25 page
On the generalized Kesten--McKay distributions
We examine the properties of distributions with the density of the form: where are
some parameters and a suitable constant. We find general forms of , of th moment and of th polynomial orthogonal with respect to
such measures. We also calculate Cauchy transforms of these measures. We
indicate connections of such distributions with distributions and polynomials
forming the so-called Askey--Wilson scheme. On the way, we prove several
identities concerning rational symmetric functions. Finally, we consider the
case of parameters forming conjugate pairs and give some
multivariate interpretations based on the obtained distributions at least for
the cases Comment: 14 page
Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions
We study a generalization of the Kibble-Slepian (KS) expansion formula in 3
dimensions. The generalization is obtained by replacing the Hermite polynomials
by the q-Hermite ones. If such a replacement would lead to non-negativity for
all allowed values of parameters and for all values of variables ranging over
certain Cartesian product of compact intervals then we would deal with a
generalization of the 3 dimensional Normal distribution. We show that this is
not the case. We indicate some values of the parameters and some compact set in
R^{3} of positive measure, such that the values of the extension of KS formula
are on this set negative. Nevertheless we indicate other applications of so
generalized KS formula. Namely we use it to sum certain kernels built of the
Al-Salam-Chihara polynomials for the cases that were not considered by other
authors. One of such kernels sums up to the Askey-Wilson density disclosing its
new, interesting properties. In particular we are able to obtain a
generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we
indicate some new interesting properties of the Askey-Wilson polynomials with
complex parameters. We also pose several open questions
A few remarks on orthogonal polynomials
Knowing a sequence of moments of a given, infinitely supported, distribution
we obtain quickly: coefficients of the power series expansion of monic
polynomials that are orthogonal with respect
to this distribution, coefficients of expansion of in the series of
, two sequences of coefficients of the 3-term recurrence of
the family of , the so called "linearization
coefficients" i.e. coefficients of expansion of in the series of
\newline Conversely, assuming knowledge of the two
sequences of coefficients of the 3-term recurrence of a given family of
orthogonal polynomials we express with
their help: coefficients of the power series expansion of , coefficients
of expansion of in the series of moments of the
distribution that makes polynomials
orthogonal. \newline Further having two different families of orthogonal
polynomials and and knowing for each of them sequences of the 3-term recurrences,
we give sequence of the so called "connection coefficients" between these two
families of polynomials. That is coefficients of the expansions of in
the series of \newline We are able to do all this due to
special approach in which we treat vector of orthogonal polynomials as a linear transformation of the
vector by some lower triangular matrix Comment: 18 page
q-Gaussian distributions. Simplifications and simulations
We present some properties of measures (q-Gaussian) that orthogonalize the
set of q-Hermite polynomials. We also present an algorithm for simulating
i.i.d. sequences of random variables having q-Gaussian distribution.Comment: 13 page
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