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1

The local structure of q-Gaussian processes

Bryc W., Wang Y.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 2

335--352

EN

The local structure of q-Ornstein-Uhlenbeck process and q-Brownian motion are investigated for all q ϵ (−1, 1). These are classical Markov processes that arose from the study of noncommutative probability. These processes have discontinuous sample paths, and the local small jumps are characterized by tangent processes. It is shown that, for all q ϵ (−1, 1), the tangent processes in the interior of the state space are scaled Cauchy processes possibly with drifts. The tangent processes at the boundary of the state space are also computed, but they are not well-known processes in classical probability theory. Instead, they can be associated with the free 1/2-stable law, a well-known distribution in free probability, via Biane’s construction.

2

Fock space realizations of some classical Markov processes

Bryc W., Ejsmont W.

Probability and Mathematical Statistics

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2013

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Vol. 33, Fasc. 2

301--313

EN

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].

3

Professor Agnieszka Plucińska's paths in and around probability

Bryc W., Wesołowski J.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

215-225

4

On integrability of quadratic harnesses

Bryc W.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

275-282

EN

We investigate integrability properties of processes with linear regressions and quadratic conditional variances. We establish the right order of dependence of which moments are finite on the parameter (…) defined below, raising the question of determining the optimal constant.

5

Free exponential families as kernel families

Bryc W.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

657-672

EN

Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from the approach analogous to the definition of exponential families by using the Cauchy-Stieltjes kernel 1/(1 - Qx) instead of the exponential kernel exp(Qx). We use this approach to re-derive some known results and to study further similarities with exponential families and reproductive exponential models.