Free martingale polynomials

By Michael Anshelevich. 2001

83 citations recorded by the source corpus.

Abstract In this paper we investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Levy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dc50ce595b40a96b0b0e9b10fc13614cac7e156f

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In Applied Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution (2007, background): “Now, it is not hard to obtain an explicit formula for the Cauchy transform of μb,c – see the displayed equations in Theorem 4 of [2] (the parameters “a, t” from those equations have to be suitably set, in order to match the b, c from the above Equation (4.”
Free Probability and Random Matrices (2014, background): “2) There exists a huge body of results around infinitely divisible and stable laws in the free sense; see, for example, [9, 8, 11, 22, 29, 30, 31, 32, 54, 70, 97, 199].”
Askey--Wilson polynomials, quadratic harnesses and martingales (2008, background): “als for stochastic processes have been studied by a number of authors, see [5, 13, 18, 22, 24, 25, 27, 28]. Orthogonal martingale polynomials play also a prominent role in non-commutative probability [1, 2], and can serve as a connection to the so called “classical versions” of non-commutative proce”
Conditional moments of q-Meixner processes (2004, background): “They areM arkov,and arisefrom thenon-com m utative constructions,atleastforthose valuesofthe param eterswhen such constructions are known.”
On a class of free Lévy laws related to a regression problem (2004, methodology): “Anshelevich [3, Remark 6] points out the q-Meixner systems of polynomials that interpolate between the classical case q = 1 and the free case q = 0.”
Free analysis questions I: duality transform for the coalgebra of ∂X:B (2004, background): “The dual algebra in this combinatorial context is related to sequences of classical polynomials and umbral calculus (see [1] for a recent free probability extension).”