## Conditional moments of q-Meixner processes (2004)

Citations: | 9 - 5 self |

### BibTeX

@TECHREPORT{Weso̷lowski04conditionalmoments,

author = {Jacek Weso̷lowski},

title = {Conditional moments of q-Meixner processes},

institution = {},

year = {2004}

}

### OpenURL

### Abstract

Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes. 1.

### Citations

603 | An Introduction to Orthogonal Polynomials - Chihara - 1978 |

201 | Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials - Askey, Wilson - 1985 |

183 | Analytic Theory of Continued Fractions - Wall - 1948 |

181 | The Problem of Moments - Shohat, Tamarkin - 1943 |

143 | Orthogonal Polynomials of Several Variables - Dunkl, Xu - 2001 |

73 | q-Gaussian processes: non-commutative and classical aspects
- Bożejko, Kümmerer, et al.
- 1997
(Show Context)
Citation Context ...ends [32, Theorem 2] to the more general quadratic conditional variances. Similar analysis of stationary sequences in [15] yields the classical versions of the non-commutative q-Gaussian processes of =-=[11]-=-. Further contributions to the stationary case can be found in [21]. Stochastic processes with linear conditional expectations and quadratic conditional variances turn out to depend on three numerical... |

63 | An Introduction to Orthogonal Polynomials,’’ Gordon 6 - Chihara - 1978 |

56 |
Recurrence relations, continued fractions and orthogonal polynomials
- Askey, Ismail
- 1984
(Show Context)
Citation Context ...m 3.5 shows that Xt is Markov with uniquely determined transition probabilities. Formulas (54) and (55) give the distribution which orthogonalizes the corresponding Al-Salam– Chihara polynomials, see =-=[8]-=-. □ 4.2. Lévy processes with quadratic conditional variance. A special choice of the coefficients in (6) casts the conditional variance as a quadratic function of the increments of the process, (58) V... |

54 | Convolution and limit theorems for conditionally free random variables
- Bożejko, Leinert, et al.
- 1996
(Show Context)
Citation Context ... − √ (1 − zθ) 2 − 4z2τ . 2zτ The free Brownian and free Poisson processes have been studied in considerable detail, see [29] and the references therein. Symmetric free Meixner distribution appears in =-=[12]-=-, and in [13, Theorem 3]. All five distributions occur in Anshelevich [5, Theorem 4]; Anshelevich also points out that the correspondence between the classical and free Levy processes based on the val... |

48 |
Processes with free increments
- Biane
- 1998
(Show Context)
Citation Context ...nomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes. 1. Introduction 1.1. Motivation. It has been known since the work of Biane =-=[10]-=- that every noncommutative process with free increments gives rise to a classical Markov process, whose transition probabilities ”realize” the non-commutative free convolution of the corresponding mea... |

44 | Stochastic Processes and Orthogonal Polynomials - Schoutens - 2000 |

34 | On a class of free Lévy laws related to a regression problem - Bozejko, Bryc |

30 | Free martingale polynomials
- Anshelevich
(Show Context)
Citation Context ...esses arise as the classical version from the q-Poisson process defined in [3, Def. 6.16]. When q = 0 the q-Meixner processes are related to the class of free Lévy processes considered by Anshelevich =-=[4]-=-. The reasons why these special cases of q-Meixner processes should arise from the Fock space constructions are not clear to us. It is not known whether the generic q-Meixner process arises as a class... |

20 |
Lectures on free probability theory, in “Lectures on probability theory and statistics
- Voiculescu
- 2000
(Show Context)
Citation Context ... known, and can be easily seen from the corresponding R-series RXs(z) = s 1 − zθ − √ (1 − zθ) 2 − 4z2τ . 2zτ The free Brownian and free Poisson processes have been studied in considerable detail, see =-=[29]-=- and the references therein. Symmetric free Meixner distribution appears in [12], and in [13, Theorem 3]. All five distributions occur in Anshelevich [5, Theorem 4]; Anshelevich also points out that t... |

19 |
Partition-dependent stochastic measures and q-deformed cumulants
- ANSHELEVICH
- 2001
(Show Context)
Citation Context ... less constraining and easier to handle. Non-commutative processes with free increments can be thought as a special case corresponding to the value q = 0 of the more general class of q-Lévy processes =-=[3]-=-, [7]. Markov processes are known to arise in this more general setting in two important cases: Bo˙zejko, Kümmerer, and Speicher, give explicit Markov transition probabilities for the q-Brownian motio... |

19 | The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory - Saitoh |

17 | Appell polynomials and their relatives
- Anshelevich
(Show Context)
Citation Context ...oisson process. Other q-Lévy processes are still not well understood, so it is not known whether Markov processes arise in the general case; for indications that Markov property may perhaps fail, see =-=[4]-=-. This paper arose as an attempt to better understand the emergence of related Markov processes from probabilistic assumptions. We define our class of processes by assuming that the first two conditio... |

16 |
Convolutions of orthonormal polynomials
- Al-Salam, Chihara
- 1976
(Show Context)
Citation Context ...rocesses are stationary, but this property is not inherited by the classical version. We use the standard notation 3. q-Meixner Markov processes [n]q = 1 + q + · · · + q n−1 , [n]q! = [ ] n = k q [1]q=-=[2]-=-q . . . [n]q, [n]q! [n − k]q![k]q! , with the usual conventions [0]q = 0, [0]q! = 1. For fixed real parameters x, s, t, q, θ, τ, define the polynomials Qn in variable y by the three step recurrence (3... |

14 | ψ-independent and symmetrized white noises - Bożejko, Speicher |

14 | Stationary random fields with linear regressions
- Bryc
(Show Context)
Citation Context ...te: Wiener, Poisson, Pascal, Gamma, and Meixner. Our main result, Theorem 3.5, extends [33, Theorem 2] to the more general quadratic conditional variances. Similar analysis of stationary sequences in =-=[16]-=- yields the classical versions of the non-commutative q-Gaussian processes of [12]. Further contributions to the stationary case can be found in [22]. Stochastic processes with linear conditional expe... |

12 | ψ-independent and symmetrized white noises, Quantum probability & related topics - Bo˙zejko, Speicher - 1991 |

12 |
Stochastic processes with linear conditional expectation and quadratic conditional variance
- Weso̷lowski
- 1993
(Show Context)
Citation Context ...nt #INT-0332062 and by the C.P. Taft Memorial Fund. 12 W̷LODZIMIERZ BRYC AND JACEK WESO̷LOWSKI sequences [16], L2-differentiable processes [28], Poisson process [14], Gamma process [31]. Weso̷lowski =-=[32]-=- unified several partial results, identifying the general quadratic conditional variance problem which characterizes the five Lévy processes of interest in this note: Wiener, Poisson, Pascal, Gamma, a... |

11 |
The classical moment problem ,Oliver and
- Akhiezer
- 1965
(Show Context)
Citation Context ...ve processes are stationary, but this property is not inherited by the classical version. We use the standard notation 3. q-Meixner Markov processes [n]q = 1 + q + · · · + q n−1 , [n]q! = [ ] n = k q =-=[1]-=-q[2]q . . . [n]q, [n]q! [n − k]q![k]q! , with the usual conventions [0]q = 0, [0]q! = 1. For fixed real parameters x, s, t, q, θ, τ, define the polynomials Qn in variable y by the three step recurrenc... |

11 |
Some basic theorems on harnesses. Stochastic analysis (a tribute to the memory of Rollo
- Williams
- 1973
(Show Context)
Citation Context ...it as u → ∞ we see that (4) E(Xt|F≤s) = Xs for 0 ≤ s ≤ t. Similarly, taking s = 0 in (2) we get (5) E(Xt|F≥u) = t u Xu. Processes which satisfy condition (2) are sometimes called harnesses, see [20], =-=[33]-=-. We assume in addition that the conditional variance of Xt given F≤s ∨F≥u is given by a quadratic expression in Xs, Xu. Recall that the conditional variance of X with respect to a σ-field F is define... |

10 | Probabilistic aspects of AlSalam-Chihara polynomials - Bryc, Matysiak, et al. |

10 | q-Lévy processes
- Anshelevich
- 2004
(Show Context)
Citation Context ... constraining and easier to handle. Non-commutative processes with free increments can be thought as a special case corresponding to the value q = 0 of the more general class of q-Lévy processes [3], =-=[6]-=-. Markov processes are known to arise in this more general setting in two important cases: Bo˙zejko, Kümmerer, and Speicher, give explicit Markov transition probabilities for the q-Brownian motion, se... |

9 |
A few remarks on Bryc’s paper on random fields with linear regressions
- Matysiak, Szabłowski
(Show Context)
Citation Context ...ances. Similar analysis of stationary sequences in [15] yields the classical versions of the non-commutative q-Gaussian processes of [11]. Further contributions to the stationary case can be found in =-=[21]-=-. Stochastic processes with linear conditional expectations and quadratic conditional variances turn out to depend on three numerical parameters −∞ < θ < ∞, τ ≥ 0, and −1 ≤ q ≤ 1. They are Markov, and... |

6 | Linearization coefficients for orthogonal polynomials using stochastic processes”, Ann
- Anshelevich
(Show Context)
Citation Context .... (v) q-Meixner type processes: θ 2 < 4τ. Some of these generalizations have already been studied in the non-commutative probability; for the q-Brownian motion see [12], for the q-Poisson process see =-=[7]-=-, [23], [25], and the references therein. Anshelevich [4, Remark 6] states a recurrence which is equivalent to (38) for s = 0, x = 0; the latter, written as (44), plays the role in our proof of Theore... |

5 |
A characterization of infinite gaussian sequences by conditional moments
- Bryc, Plucińska
- 1985
(Show Context)
Citation Context ...00 Mathematics Subject Classification. Primary: 60J25. Research partially supported by NSF grant #INT-0332062 and by the C.P. Taft Memorial Fund. 12 W̷LODZIMIERZ BRYC AND JACEK WESO̷LOWSKI sequences =-=[16]-=-, L2-differentiable processes [28], Poisson process [14], Gamma process [31]. Weso̷lowski [32] unified several partial results, identifying the general quadratic conditional variance problem which cha... |

4 |
martingale polynomials
- Free
(Show Context)
Citation Context ...esses arise as the classical version from the q-Poisson process defined in [3, Def. 6.16]. When q = 0 the q-Meixner processes are related to the class of free Lévy processes considered by Anshelevich =-=[5]-=-. The reasons why these special cases of q-Meixner processes should arise from the Fock space constructions are not clear to us. It is not known whether the generic q-Meixner process arises as a class... |

4 |
On a stochastic process determined by the conditional expectation and the conditional variance, Stochastics 10
- Plucińska
- 1983
(Show Context)
Citation Context ...neral processes the assumption of continuity of trajectories fails, so we replace it by conditioning with respect not only to the past, but also to the future. This approach originated with Plucińska =-=[23]-=- who proved that processes with linear conditional expectations and constant conditional variances are Gaussian. Subsequent papers covered discrete Gaussian Date: February 17, 2004. 2000 Mathematics S... |

4 |
Can the first two conditional moments identify a mean square diffentiable process
- Szablowski
- 1989
(Show Context)
Citation Context ...ion. Primary: 60J25. Research partially supported by NSF grant #INT-0332062 and by the C.P. Taft Memorial Fund. 12 W̷LODZIMIERZ BRYC AND JACEK WESO̷LOWSKI sequences [16], L2-differentiable processes =-=[28]-=-, Poisson process [14], Gamma process [31]. Weso̷lowski [32] unified several partial results, identifying the general quadratic conditional variance problem which characterizes the five Lévy processes... |

3 |
Spectra of Hamiltonians with generalized single-site dynamical disorder
- Neu, Speicher
- 1994
(Show Context)
Citation Context ... q-Meixner type processes: θ 2 < 4τ. Some of these generalizations have already been studied in the non-commutative probability; for the q-Brownian motion see [11], for the q-Poisson process see [6], =-=[22]-=-, [24], and the references therein. Anshelevich [5, Remark 6] states a recurrence which is equivalent to (38) for s = 0, x = 0; the latter, written as (44), plays the role in our proof of Theorem 3.5.... |

2 |
A characterization of the Poisson process by conditional moments
- Bryc
- 1987
(Show Context)
Citation Context ...esearch partially supported by NSF grant #INT-0332062 and by the C.P. Taft Memorial Fund. 12 W̷LODZIMIERZ BRYC AND JACEK WESO̷LOWSKI sequences [16], L2-differentiable processes [28], Poisson process =-=[14]-=-, Gamma process [31]. Weso̷lowski [32] unified several partial results, identifying the general quadratic conditional variance problem which characterizes the five Lévy processes of interest in this n... |

2 |
random fields with linear regressions
- Stationary
(Show Context)
Citation Context ...te: Wiener, Poisson, Pascal, Gamma, and Meixner. Our main result, Theorem 3.5, extends [32, Theorem 2] to the more general quadratic conditional variances. Similar analysis of stationary sequences in =-=[15]-=- yields the classical versions of the non-commutative q-Gaussian processes of [11]. Further contributions to the stationary case can be found in [21]. Stochastic processes with linear conditional expe... |

2 |
The q-deformed Poisson random variables on the q-Fock space
- Saitoh, Yoshida
(Show Context)
Citation Context ...xner type processes: θ 2 < 4τ. Some of these generalizations have already been studied in the non-commutative probability; for the q-Brownian motion see [11], for the q-Poisson process see [6], [22], =-=[24]-=-, and the references therein. Anshelevich [5, Remark 6] states a recurrence which is equivalent to (38) for s = 0, x = 0; the latter, written as (44), plays the role in our proof of Theorem 3.5. Howev... |

2 |
A characterization of the gamma process by conditional moments
- Weso̷lowski
- 1989
(Show Context)
Citation Context ...pported by NSF grant #INT-0332062 and by the C.P. Taft Memorial Fund. 12 W̷LODZIMIERZ BRYC AND JACEK WESO̷LOWSKI sequences [16], L2-differentiable processes [28], Poisson process [14], Gamma process =-=[31]-=-. Weso̷lowski [32] unified several partial results, identifying the general quadratic conditional variance problem which characterizes the five Lévy processes of interest in this note: Wiener, Poisson... |