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Free martingale polynomials. (English) Zbl 1033.46050

Summary: We investigate the properties of free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy 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.

MSC:

46L54 Free probability and free operator algebras
05A40 Umbral calculus
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
60G44 Martingales with continuous parameter
47H20 Semigroups of nonlinear operators
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[1] Anshelevich, M.: The linearization of the central limit operator in free probability theory. Probab. theory related fields 115, No. 3, 401-416 (1999) · Zbl 0983.46051
[2] Anshelevich, M.: Free stochastic measures via noncrossing partitions. Adv. math. 155, No. 1, 154-179 (2000) · Zbl 0978.46041
[3] Anshelevich, M.: Partition-dependent stochastic measures and q-deformed cumulants. Doc. math. 6, 343-384 (2001) · Zbl 1010.46064
[4] Anshelevich, M.: Itô formula for free stochastic integrals. J. funct. Anal. 188, No. 1, 292-315 (2002) · Zbl 0998.46031
[5] Askey, R.; Wilson, J.: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. amer. Math. soc. 54, No. 319, iv+55 (1985) · Zbl 0572.33012
[6] Akiyama, M.; Yoshida, H.: The orthogonal polynomials for a linear sum of a free family of projections. Infinite dimensional anal. Quantum probab. Related top. 2, No. 4, 627-643 (1999) · Zbl 1043.46509
[7] P. Biane, Free Brownian motion, free stochastic calculus and random matrices, in: Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun., Vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 1–19. · Zbl 0873.60056
[8] Biane, P.: Processes with free increments. Math. Z. 227, No. 1, 143-174 (1998) · Zbl 0902.60060
[9] Bo\.Zejko, M.; Kümmerer, B.; Speicher, R.: Q-Gaussian processesnon-commutative and classical aspects. Comm. math. Phys. 185, No. 1, 129-154 (1997) · Zbl 0873.60087
[10] Bercovici, H.; Pata, V.: Stable laws and domains of attraction in free probability theory. Ann. of math. (2) 149, No. 3, 1023-1060 (1999) · Zbl 0945.46046
[11] Cabanal-Duvillard, T.: Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. inst. H. Poincaré probab. Statist. 37, No. 3, 373-402 (2001) · Zbl 1016.15020
[12] Cohen, J. M.; Trenholme, A. R.: Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis. J. funct. Anal. 59, No. 2, 175-184 (1984) · Zbl 0549.43002
[13] Di Bucchianico, A.; Loeb, D.: A selected survey of umbral calculus. Electron. J. Combin. 2 (1995) · Zbl 0851.05012
[14] Feinsilver, P.; Schott, R.: Representations and probability theory. Algebraic structures and operator calculus (1993) · Zbl 0782.60015
[15] Haagerup, U.; Thorbjørnsen, S.: Random matrices and K-theory for exact C*-algebras. Doc. math. 4, 341-450 (1999) · Zbl 0933.46051
[16] Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke math. J. 91, No. 1, 151-204 (1998) · Zbl 1039.82504
[17] Joni, S. A.; Rota, G. -C.: Coalgebras and bialgebras in combinatorics. Stud. appl. Math. 61, No. 2, 93-139 (1979) · Zbl 0471.05020
[18] Lai, Tze Leung: Martingales and boundary crossing probabilities for Markov processes. Ann. probability 2, 1152-1167 (1974) · Zbl 0326.60054
[19] Lehner, F.: On the computation of spectra in free probability. J. funct. Anal. 183, No. 2, 451-471 (2001) · Zbl 1020.46021
[20] Jr., H. P. Mckean: Stochastic integrals. (1969) · Zbl 0191.46603
[21] Rota, G. -C.; Kahaner, D.; Odlyzko, A.: On the foundations of combinatorial theory. VIII. finite operator calculus. J. math. Anal. appl. 42, 684-760 (1973) · Zbl 0267.05004
[22] G.-C. Rota (Ed.), Finite Operator Calculus, Academic Press Inc., New York, 1975.
[23] Schoutens, W.: Stochastic processes and orthogonal polynomials. (2000) · Zbl 0960.60076
[24] Speicher, R.: Free probability theory and non-crossing partitions. Sém. lothar. Combin. 39 (1997) · Zbl 0887.46036
[25] Saitoh, N.; Yoshida, H.: A q-deformed Poisson distribution based on orthogonal polynomials. J. phys. A 33, No. 7, 1435-1444 (2000) · Zbl 0955.33013
[26] Saitoh, N.; Yoshida, H.: The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. math. Statist. 21, No. 1, 159-170 (2001) · Zbl 1020.46019
[27] Szegö, G.: Über die entwicklung einer willkurlichen funktion nach den polynomen eines orthogonalsystems. Math. zeit. 12, 61-94 (1922) · JFM 48.0378.03
[28] D.V. Voiculescu, K.J. Dykema, A. Nica, A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, Free Random Variables, CRM Monograph Series, Vol. 1, American Mathematical Society, Providence, RI, 1992. · Zbl 0795.46049
[29] Voiculescu, D.: Dual algebraic structures on operator algebras related to free products. J. oper. Theory 17, No. 1, 85-98 (1987) · Zbl 0656.46058
[30] D. Voiculescu, The coalgebra of the free difference quotient and free probability, Internat. Math. Res. Notices (2) (2000) 79–106. · Zbl 0952.46038
[31] D. Voiculescu, Lectures on Free Probability Theory, Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Springer, Berlin, 2000, pp. 279–349. · Zbl 1015.46037
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