Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

We study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as “Poisson type” limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.

Matching the distributions of the marginals and the sums for the Meixner class

Для заданного множества независимых случайных величин $X_1,…,X_d$, принадлежащих классу Мейкснера, найдены случайные величины $Y_1,…,Y_d$, для которых маргинальные распределения и распределения сумм совпадают: $Y_i\stackrel{d}{=} X_i$ и $\sum_iY_i\stackrel{d}{=}\sum_iX_i$. В работе дается полная характеризация случайных величин $Y_1,…,Y_d$ и предлагаются примеры построения с помощью конечных среднеквадратичных разложений.

Meixner class of orthogonal polynomials of a non-commutative monotone Lévy noise

Let [Formula: see text] denote a non-commutative monotone Lévy process. Let [Formula: see text] denote the corresponding monotone Lévy noise, i.e. formally [Formula: see text]. A continuous polynomial of [Formula: see text] is an element of the corresponding non-commutative [Formula: see text]-space [Formula: see text] that has the form [Formula: see text], where [Formula: see text]. We denote by [Formula: see text] the space of all continuous polynomials of [Formula: see text]. For [Formula: see text], the orthogonal polynomial [Formula: see text] is defined as the orthogonal projection of the monomial [Formula: see text] onto the subspace of [Formula: see text] that is orthogonal to all continuous polynomials of [Formula: see text] of order [Formula: see text]. We denote by [Formula: see text] the linear span of the orthogonal polynomials. Each orthogonal polynomial [Formula: see text] depends only on the restriction of the function [Formula: see text] to the set [Formula: see text]. The orthogonal polynomials allow us to construct a unitary operator [Formula: see text], where [Formula: see text] is an extended monotone Fock space. Thus, we may think of the monotone noise [Formula: see text] as a distribution of linear operators acting in [Formula: see text]. We say that the orthogonal polynomials belong to the Meixner class if [Formula: see text]. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: [Formula: see text] and [Formula: see text]. In this case, the monotone Lévy noise has the representation [Formula: see text]. Here, [Formula: see text] and [Formula: see text] are the (formal) creation and annihilation operators at [Formula: see text] acting in [Formula: see text].

ON STOCHASTIC DOMINANCE OF NILPOTENT OPERATORS

In this paper, motivated by Nica and Speicher [Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, 2006)] and Kubo and Kuo [MRM-factors for the probability measures in the Meixner class, Infin. Dimens. Anal. Quantum Probab. Relat. Top.13 (2010) 525–550], we characterize a particular nilpotent case of a truncated forward shift operator by applying the technique of the random walks with repeated reflections and associated renewal equations. We also establish a stochastic order relationship by applying the crossing criteria.

MRM-FACTORS FOR THE PROBABILITY MEASURES IN THE MEIXNER CLASS

It is known that the gamma distribution γκ is MRM-applicable for h(x) = ex and for some hypergeometric functions also. We are interested in the problem to determine all possible MRM-factors of probability measures which are MRM-applicable for ex. We may say that the measures are in Meixner class. Such typical measures are Gaussian, Poisson, gamma, negative binomial and Meixner distributions and others are obtained from their modifications by affine transforms. We will give the complete list of MRM-factors different from ex up to trivial deformation: (1) [Formula: see text] for gamma distribution γκ. (2) [Formula: see text] for gamma distribution γκ. (3) [Formula: see text] for standard Gaussian distribution. (4) [Formula: see text] for shifted negative binomial distribution. σβ NegBin (κ,p) with κ = 2, β = 1, for Meixner distribution Mκ,η with κ = 2 and for gamma distribution γκ with κ = 2, which is a special case of (2) with c = 1.

LOWERING AND RAISING OPERATORS FOR THE FREE MEIXNER CLASS OF ORTHOGONAL POLYNOMIALS

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.