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New interpolations between classical and free Gaussian processes

Buchholz A.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

181--189

In this paper we present the base of a general technique to derive new positive definite functions on pairings from already known ones. To describe this technique we use two concrete applications. The first one refers to the function depending on the number of connected components, the second one to the function depending on the number of crossings. In the first application we get a new family of functions identifying nontrivial connected components.

Fock space realizations of some classical Markov processes

Bryc W., Ejsmont W.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

301--313

We define a pair of non-commutative processes on a perturbed Fock space. Both processes have the same univariate distributions and satisfy a weak form of the polynomial martingale property. The processes give two non-equivalent Fock-space realizations of the same classical Markov process: the two-parameter bi-Poisson processes introduced in [12], and constructed in [13].

A simple proof of the classification theorem for positive natural products

Muraki N.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

315--326

A simplification of the proof of the classification theorem for natural notions of stochastic independence is given. This simplification is made possible after adding the positivity condition to the algebraic axioms for a (non-symmetric) universal product (i.e. a natural product). Indeed, this simplification is nothing but a simplification, under the positivity, of the proof of the claim that, for any natural product, the ‘wrong-ordered’ coefficients all vanish in the expansion form. The known proof of this claim involves a cumbersome process of solving a system of quadratic equations in 102 unknowns, but in our new proof under the positivity we can avoid such a process.