Free Stochastic Measures via Noncrossing Partitions
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We consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an n-dimensional space. In this free case, one can restrict the analysis to only the noncrossing diagonals. We give definitions of what free multiple stochastic measures are, and calculate them for the free Poisson and free compound Poisson processes. We also derive general combinatorial Itô-type relationships between free stochastic measures of different orders. These allow us to calculate, for example, free Poisson-Charlier polynomials, which are the orthogonal polynomials with respect to the free Poisson measure.
23 pages, AMS-LaTeX2e; statement of Proposition 3 corrected
- University of California, Berkeley United States
free compound Poisson process, Mathematics(all), Probability (math.PR), Mathematics - Operator Algebras, Itô product formula, Noncommutative probability and statistics, free Kailath-Segall formula, 46L50 (Primary) 60J30, 60G (Secondary), noncrossing partition, Free probability and free operator algebras, free stochastic measure, free Poisson process, Poisson-Charlier polynomials, FOS: Mathematics, Operator Algebras (math.OA), Mathematics - Probability
free compound Poisson process, Mathematics(all), Probability (math.PR), Mathematics - Operator Algebras, Itô product formula, Noncommutative probability and statistics, free Kailath-Segall formula, 46L50 (Primary) 60J30, 60G (Secondary), noncrossing partition, Free probability and free operator algebras, free stochastic measure, free Poisson process, Poisson-Charlier polynomials, FOS: Mathematics, Operator Algebras (math.OA), Mathematics - Probability
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