33 citations · 182 across the 41 of their papers we have counts for
20 papers · 1 filter
The conditional expectation of the product of the first Hermite polynomials in a multivariate normal distribution with respect to the -th variable. A fresh perspective on the Kibble-Slepian formula
Paweł J. Szabłowski
We calculate the conditional expectation of given if random vector has multivariate normal distribution…
Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths
Paweł J. Szabłowski
We are studying stationary random processes with conditional polynomial moments that allow a continuous path modification. Processes with continuous path modification, are importan…
Polynomial identities that involve binomial coefficients, double and rising factorials and their probabilistic interpretations and proofs
Paweł J. Szabłowski
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial co…
Moment sequences of Beta distribution
Paweł J. Szabłowski
We recall some basic properties of the Beta distribution and some of its modifications. We identified around of the moment sequences of Beta distributions as important integer…
On positivity of orthogonal series and its applications in probability
Paweł J. Szabłowski
We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analys…
On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials
Paweł J. Szabłowski
We study properties of compactly supported, 4 parameter \newline family of continuous type 3 dimensional distributions, that have th…
Extracting information from random data. Applications of laws of large numbers in technical sciences and statistics
Paweł J. Szabłowski
We formulate conditions for convergence of Laws of Large Numbers and show its links with of the parts of mathematical analysis such as summation theory, convergence of orthogonal s…
On the generalized Kesten--McKay distributions
Paweł J. Szabłowski
We examine the properties of distributions with the density of the form: where $c,a_{1},\ldots…
Moments of q-Normal and conditional q-Normal distributions
Paweł J. Szabłowski
We calculate moments and moment generating functions of two distributions: the so called Normal and the so called conditional Normal distributions. These distributions gene…
Markov processes, polynomial martingales and orthogonal polynomials
Paweł J. Szabłowski
We study general properties for the family of stochastic processes with polynomial regression property, that is that every conditional moment of the process is a polynomial. It tur…
On moments of Cantor and related distributions
Paweł J. Szabłowski
We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having d…
On stationary Markov processes with polynomial conditional moments
Paweł J. Szabłowski
We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say is a polynomial of degree a…
Lévy processes, martingales, reversed martingales and orthogonal polynomials
Paweł J. Szabłowski
We study class of Lévy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left\{ M_{n}(X_{t},t),\mathcal{F}_{\leq…
On Markov processes with polynomials conditional moments
Paweł J. Szabłowski
We study properties of a subclass of Markov processes that have all moments that are continuous functions of the time parameter and more importantly are characterized by the proper…
A few remarks on quadratic harnesses
Paweł J. Szabłowski
We analyze and partially solve system of recurrences that can be derived from the properties of martingale orthogonal polynomials that characterize quadratic harnesses (QH). We als…
Multidimensional q-Normal and related distributions - Markov case
Paweł J. Szabłowski
We define and study distributions in R^{d} that we call q-Normal. For q=1 they are really multidimensional Normal, for q\in(-1,1) they have densities, compact support and many prop…
Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials
Paweł J. Szabłowski
We prove the existence of stationary random fields with linear regressions for and thus close an open question posed by W. Bryc et al.. We prove this result by describing a d…
Bryc's random fields: the existence and distributions analysis
Wojciech Matysiak, Paweł J. Szabłowski
We examine problem of existence of stationary random fields with linear regressions and quadratic conditional variances, introduced by Bryc in "Stationary random fields with linear…
Generalized stationary random fields with linear regressions - an operator approach
Wojciech Matysiak, Paweł J. Szabłowski
Existence, -stationarity and linearity of conditional expectations $\wwo{X_k}{...,X_{k-2},X_{k-1}}$ of square integrable random sequences …
Wiener ( Ornstein-Uhlenbeck processes. A generalization of known processes
Paweł J. Szabłowski
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. Altho…