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Quadratic harnesses, $$q$$-commutations, and orthogonal martingale polynomials. (English) Zbl 1129.60068
40 years ago Hammersley introduced the concept of harnesses on $$\mathbb{R}^n$$ as probabilistic models of longrange misorientation in the crystalline structure of metals. An important subclass are the so-called quadratic harnesses. The main focus of the paper is on uniqueness and properties. More precisely, the authors show that quadratic harnesses are described via five numerical constants, which, under certain integrability conditions, in fact, determine the process. Another topic concerns properties of martingale polynomials associated with quadratic harnesses with finite moments of all orders processes. Some glimps into ongoing further work and future possible work are also given.

##### MSC:
 60J25 Continuous-time Markov processes on general state spaces 46L53 Noncommutative probability and statistics
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