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Semigroups of Distributions with Linear Jacobi Parameters

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Abstract

We show that a convolution semigroup {μ t } of measures has Jacobi parameters polynomial in the convolution parameter t if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the two-state free convolution. This class of two-state free convolution semigroups has not been considered explicitly before. We show that it also has Meixner-type properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a Laha–Lukacs-type characterization, and is related to the q=0 case of quadratic harnesses.

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References

  1. Accardi, L., Bożejko, M.: Interacting Fock spaces and Gaussianization of probability measures. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(4), 663–670 (1998). MR1665281 (2000d:60158)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anshelevich, M.: Partition-dependent stochastic measures and q-deformed cumulants. Doc. Math. 6, 343–384 (2001) (electronic). MR1871667 (2004k:46107)

    MathSciNet  MATH  Google Scholar 

  3. Anshelevich, M.: Free martingale polynomials. J. Funct. Anal. 201(1), 228–261 (2003). MR1986160 (2004f:46079)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anshelevich, M.: Orthogonal polynomials with a resolvent-type generating function. Trans. Am. Math. Soc. 360(8), 4125–4143 (2008). MR2395166 (2009g:33012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Anshelevich, M.: Appell polynomials and their relatives. II. Boolean theory. Indiana Univ. Math. J. 58(2), 929–968 (2009). MR2514394 (2010d:46090)

    Article  MathSciNet  MATH  Google Scholar 

  6. Belinschi, S.T.: C-free convolution for measures with unbounded support. In: Von Neumann Algebras in Sibiu, Theta Ser. Adv. Math., vol. 10, Theta, Bucharest, pp. 1–7 (2008). MR2512322

    Google Scholar 

  7. Bożejko, M., Bryc, W.: On a class of free Lévy laws related to a regression problem. J. Funct. Anal. 236(1), 59–77 (2006). MR2227129 (2007a:46071)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bożejko, M., Bryc, W.: A quadratic regression problem for two-state algebras with an application to the central limit theorem. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(2), 231–249 (2009). MR2541395

    Article  MathSciNet  MATH  Google Scholar 

  9. Bożejko, M., Kümmerer, B., Speicher, R.: q-Gaussian processes: non-commutative and classical aspects. Commun. Math. Phys. 185(1), 129–154 (1997). MR1463036 (98h:81053)

    Article  MATH  Google Scholar 

  10. Bożejko, M., Leinert, M., Speicher, R.: Convolution and limit theorems for conditionally free random variables. Pac. J. Math. 175(2), 357–388 (1996). MR1432836 (98j:46069)

    MATH  Google Scholar 

  11. Bożejko, M., Lytvynov, E.: Meixner class of non-commutative generalized stochastic processes with freely independent values. I. A characterization. Commun. Math. Phys. 292(1), 99–129 (2009). MR2540072

    Article  MATH  Google Scholar 

  12. Bożejko, M., Wysoczański, J.: Remarks on t-transformations of measures and convolutions. Ann. Inst. Henri Poincaré Probab. Stat. 37(6), 737–761 (2001). MR1863276 (2002i:60005)

    Article  MATH  Google Scholar 

  13. Bryc, W.: Stationary random fields with linear regressions. Ann. Probab. 29(1), 504–519 (2001). MR1825162 (2002d:60014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bryc, W.: Free exponential families as kernel families. Demonstr. Math. 42(3), 657–672 (2009). MR2554960

    MathSciNet  Google Scholar 

  15. Bryc, W., Matysiak, W., Wesołowski, J.: Quadratic harnesses, q-commutations, and orthogonal martingale polynomials. Trans. Am. Math. Soc. 359(11), 5449–5483 (2007). MR2327037 (2008j:60176)

    Article  MATH  Google Scholar 

  16. Bryc, W., Wesołowski, J.: Conditional moments of q-Meixner processes. Probab. Theory Relat. Fields 131(3), 415–441 (2005). MR2123251 (2005k:60233)

    Article  MATH  Google Scholar 

  17. Bryc, W., Wesołowski, J.: Bi-Poisson process. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(2), 277–291 (2007). MR2337523 (2008d:60097)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bryc, W., Wesołowski, J.: Askey-Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. 38(3), 1221–1262 (2010). MR2674998 (2011d:60226)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol. 13. Gordon & Breach, New York (1978). MR0481884 (58 #1979)

    MATH  Google Scholar 

  20. Diaconis, P., Khare, K., Saloff-Coste, L.: Gibbs sampling, exponential families and orthogonal polynomials. Stat. Sci. 23(2), 151–200 (2008) with comments and a rejoinder by the authors. MR2446500

    Article  MathSciNet  Google Scholar 

  21. Dykema, K.J.: Multilinear function series and transforms in free probability theory. Adv. Math. 208(1), 351–407 (2007). MR2304321 (2008k:46193)

    Article  MathSciNet  MATH  Google Scholar 

  22. Flajolet, P.: Combinatorial aspects of continued fractions. Discrete Math. 32(2), 125–161 (1980). MR592851 (82f:05002a)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hamada, M., Konno, N., Młotkowski, W.: Orthogonal polynomials induced by discrete-time quantum walks in one dimension. Interdiscip. Inf. Sci. 15(3), 367–375 (2009). 2598327

    MATH  Google Scholar 

  24. Hasebe, T., Saigo, H.: The monotone cumulants. Ann. Inst. Henri Poincaré Probab. Stat. 47(4), 1160–1170 (2011). arXiv:0907.4896v2 [math.PR]

    Article  MathSciNet  Google Scholar 

  25. Hinz, M., Młotkowski, W.: Free cumulants of some probability measures. In: Noncommutative Harmonic Analysis with Applications to Probability. Banach Center Publ., vol. 78, pp. 165–170. Polish Acad. Sci. Inst. Math., Warsaw (2007). MR2402356 (2009f:46088)

    Chapter  Google Scholar 

  26. Hinz, M., Młotkowski, W.: Limit measures related to the conditionally free convolution. Bull. Pol. Acad. Sci., Math. 56(1), 75–81 (2008). MR2411028 (2009h:46129)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kim, D., Zeng, J.: A combinatorial formula for the linearization coefficients of general Sheffer polynomials. Eur. J. Comb. 22(3), 313–332 (2001). MR1822720 (2001k:05023)

    Article  MathSciNet  MATH  Google Scholar 

  28. Krystek, A.D., Wojakowski, Ł.J.: Associative convolutions arising from conditionally free convolution. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(3), 515–545 (2005). MR2172313 (2006g:46103)

    Article  MathSciNet  MATH  Google Scholar 

  29. Laha, R.G., Lukacs, E.: On a problem connected with quadratic regression. Biometrika 47, 335–343 (1960). MR0121922 (22 #12649)

    MathSciNet  MATH  Google Scholar 

  30. Lenczewski, R.: Decompositions of the free additive convolution. J. Funct. Anal. 246(2), 330–365 (2007). MR2321046 (2008d:28009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meixner, J.: Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. Lond. Math. Soc. 9, 6–13 (1934)

    Article  MathSciNet  Google Scholar 

  32. Młotkowski, W.: Combinatorial relation between free cumulants and Jacobi parameters. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(2), 291–306 (2009). MR2541398

    Article  MathSciNet  MATH  Google Scholar 

  33. Młotkowski, W.: Some examples of conditionally free product. In: Ojima, I. (ed.) Non-Commutative Analysis and Micro-Macro Duality. RIMS Kôkyûroku, vol. 1658, pp. 136–139 (2009)

    Google Scholar 

  34. Morris, C.N.: Natural exponential families with quadratic variance functions. Ann. Stat. 10(1), 65–80 (1982). MR642719 (83a:62037)

    Article  MATH  Google Scholar 

  35. Morris, C.N.: Natural exponential families with quadratic variance functions: statistical theory. Ann. Stat. 11(2), 515–529 (1983). MR696064 (84h:62043)

    Article  MATH  Google Scholar 

  36. Muraki, N.: Noncommutative Brownian motion in monotone Fock space. Commun. Math. Phys. 183(3), 557–570 (1997). MR1462227 (98j:81186)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge University Press, Cambridge (2006). MR2266879 (2008k:46198)

    Book  MATH  Google Scholar 

  38. Saitoh, N., Yoshida, H.: The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory. Probab. Math. Stat. 21(1), 159–170 (2001). 159–170. MR1869728 (2003e:46114)

    MathSciNet  MATH  Google Scholar 

  39. Schoutens, W.: Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, vol. 146. Springer, New York (2000). MR1761401 (2001f:60095)

    Book  MATH  Google Scholar 

  40. Shiryaev, A.N.: Probability, 2nd edn. Graduate Texts in Mathematics, vol. 95. Springer, New York (1996). Translated from the first (1980) Russian edition by R.P. Boas. MR1368405 (97c:60003)

    Google Scholar 

  41. Speicher, R.: A new example of “independence” and “white noise”. Probab. Theory Relat. Fields 84(2), 141–159 (1990). MR1030725 (90m:46116)

    Article  MathSciNet  MATH  Google Scholar 

  42. Speicher, R.: Multiplicative functions on the lattice of noncrossing partitions and free convolution. Math. Ann. 298(4), 611–628 (1994). MR1268597 (95h:05012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Speicher, R., Woroudi, R.: Boolean convolution. In: Free Probability Theory, Waterloo, ON, 1995. Fields Inst. Commun., vol. 12, pp. 267–279. Amer. Math. Soc., Providence (1997). MR1426845 (98b:46084)

    Google Scholar 

  44. Viennot, G.: Une théorie combinatoire des polynômes orthogonaux généraux. Unpublished notes, Univ. Quebec, Montreal, Que (1984)

  45. Wesołowski, J.: Stochastic processes with linear conditional expectation and quadratic conditional variance. Probab. Math. Stat. 14(1), 33–44 (1993). MR1267516 (95b:60061)

    MATH  Google Scholar 

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Acknowledgements

The paper was started during the 12th workshop on Non-commutative Harmonic Analysis with Applications to Probability at the Banach center. M.A. would like to thank the organizers for an enjoyable conference. He would also like to thank Włodek Bryc and Jacek Wesołowski for explaining their work to him. Perhaps most importantly, we thank Laura Matusevich for pointing out a missing assumption in Theorem 6, which led to Proposition 7. Finally, we are grateful to the referee for a careful reading of the paper, and numerous very useful comments and suggestions.

M.A. was supported in part by NSF grant DMS-0900935. W.M. was supported by MNiSW: N N201 364436.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Michael Anshelevich

  2. Mathematical Institute, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Wojciech Młotkowski

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  1. Michael Anshelevich
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Correspondence to Michael Anshelevich.

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Anshelevich, M., Młotkowski, W. Semigroups of Distributions with Linear Jacobi Parameters. J Theor Probab 25, 1173–1206 (2012). https://doi.org/10.1007/s10959-012-0403-x

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