Results 1  10
of
10
Conditional moments of qMeixner processes
, 2004
"... Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these proce ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a threeparameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the noncommutative generalizations of the Lévy processes. 1.
The bi  Poisson process: a quadratic harness
 Annals of Probability
"... This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a twoparameter extension of the AlSalam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The
Derived noncommutative continuous Bernoulli shifts
 In preparation
"... Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, i ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract: We introduce a noncommutative extension of TsirelsonVershik’s noises [TV98, Tsi04], called (noncommutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalarexpected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) Gstationary flows for TsirelsonVershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (noncommutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: noncommutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.
Slide 4
, 2005
"... Quadratic harness condition: = Qt,s,u(Xs, Xu), (3) E ( X 2 ∣ t Fs,u math.uc.edu/~brycw/preprint/5param/qharnesses.pdf where Qt,s,u(x, y) = At,s,ux 2 +Bt,s,uxy +Ct,s,uy 2 +Dt,s,ux+Et,s,uy +Ft,s,u (4) ..."
Abstract
 Add to MetaCart
Quadratic harness condition: = Qt,s,u(Xs, Xu), (3) E ( X 2 ∣ t Fs,u math.uc.edu/~brycw/preprint/5param/qharnesses.pdf where Qt,s,u(x, y) = At,s,ux 2 +Bt,s,uxy +Ct,s,uy 2 +Dt,s,ux+Et,s,uy +Ft,s,u (4)
q−GAUSSIAN DISTRIBUTIONS. ON CALCULUS OF MEAURES ORTHOGONALIZING qHERMITE POLYNOMIALS
, 2005
"... Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1. ..."
Abstract
 Add to MetaCart
Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.
q−GAUSSIAN DISTRIBUTIONS. SIMPLIFICATIONS AND SIMULATIONS
, 2009
"... Abstract. 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. 1. ..."
Abstract
 Add to MetaCart
Abstract. 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. 1.
q−GAUSSIAN DISTRIBUTIONS. ON CALCULUS OF MEAURES ORTHOGONALIZING qHERMITE POLYNOMIALS
, 2005
"... Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1. ..."
Abstract
 Add to MetaCart
Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.
q−GAUSSIAN DISTRIBUTIONS. ON CALCULUS OF MEAURES ORTHOGONALIZING QHERMITE POLYNOMIALS
, 2005
"... Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1. ..."
Abstract
 Add to MetaCart
Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.
q−GAUSSIAN DISTRIBUTIONS. ON CALCULUS OF MEAURES ORTHOGONALIZING QHERMITE POLYNOMIALS
, 2005
"... Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1. ..."
Abstract
 Add to MetaCart
Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.
Meixner
, 812
"... class of noncommutative generalized stochastic processes with freely independent values ..."
Abstract
 Add to MetaCart
class of noncommutative generalized stochastic processes with freely independent values