## 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.

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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martingale polynomials
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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... |

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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... |

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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.... |

3 |
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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... |

2 |
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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
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(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
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(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
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- 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... |