Multivariate Appell polynomials and the central limit theorem

  • L. Giraitis
  • D. Surgailis
Part of the Progress in Probability and Statistics book series (PRPR, volume 11)


A CLT for processes of the form L(Xt) is proved, where L(x) is a polynomial and Xt, t ∈ ℤ is a process with long range dependence. Conditions on Xt are formulated in terms of semi-invariants; they are specified for linear processes Xt. The notion of the Appell rank of L(x) plays a basic role in the CLT. Various topics related to Appell polynomials (e.g. expansions, diagram formalism for semi-invariants) are discussed.


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  1. [1]
    Avram, F., Taqqu, M.S. Non—central limit theorems and generalized powers. Cornell University 1985.Google Scholar
  2. [2]
    Berman, S.M. High level sojourns for strongly dependent Gaussian Processes. Z.Wahrsch.verw.Gebiete 50 (1979), 223–236.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    Boas, R.P., Buck, R.C. Polynomials expansions of analytic functions. Springer: New—York, 1964.CrossRefGoogle Scholar
  4. [4]
    Brillinger, D.R. Time series. Date analysis and theory. Holt, Riueart and Winston: New—York 1975.Google Scholar
  5. [5]
    Breuer, P., Major, P. Central limit theorem for non—linear functionals of Gaussian fields. J.of multivariate analysis 13 (1983), 425–441.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    Bourbaki, N. Elements de mathématique. Livre IV. Fonctions d’une variable réele. Herman: Paris,4958Google Scholar
  7. [7]
    Dobrushin, R.L., Major, P. Non—central limit theorem for non—linear functionals of Gaussian fields. Z.Wahrsch. verw. Gebiete 50 (1979), 27–52.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Feinsilver, P.J. Special functions, probability semigroups, and Hamiltonian flows. Lecture Notes in Math., 696, Springer: New—York, 1978.Google Scholar
  9. [9]
    Giraitis, L. Central limit theorem for functionals of linear processes. (Russian). 25 (1985) No 1, 43–57.MathSciNetGoogle Scholar
  10. [10]
    Giraitis, L., Surgailis, D. CLT and other limit theorems for functionals of Gaussian processes. Z.Wahrsch.verw. Gebiete (to appear).Google Scholar
  11. [11].
    Glimm, J., Jaffe, A. Quantum physics. A functional integral point of view. Springer: New—York—Berlin 1981.zbMATHGoogle Scholar
  12. [12]
    Ibragimov, I.A., Linnik, J. V. Independent and stationary sequences of random variables. Walters-Noordhoff: Groningen 1971.zbMATHGoogle Scholar
  13. [13]
    Kazmin, J.A. Appell polynomial series expantions. (Russian) Mat.Zametki 5 (1969), 509–520.MathSciNetGoogle Scholar
  14. [14]
    Krein, S.G. (ed.). Functional analysis. (in Russian). Nauka: Moscow 1972.Google Scholar
  15. [15]
    Leonov, P.P., Shiryaev, A.N. Sur le calcul des semi-invariants. Teor.Verojatn. i Primenen. 4 (1959), 342–355.Google Scholar
  16. [16]
    Lukacs, E. Characteristic functions (2 v4 ed). Griffin: London, 1970Google Scholar
  17. [17]
    Malyshev, V.A. Cluster expansions in lattice models of statistical physics and quantum field theory.(Russian). Uspeki Mat.Nauk 35 (1980), No 2, 3–53.MathSciNetGoogle Scholar
  18. [18]
    Ruiz de Chavez, J. Conpensation multiplicative et “Produits de Wick”. In: Seminaire de Probabilités XVIII Proceedings. Lecture Notes in Math., 1123, Springer: Berlin 1985, pp. 242–247.Google Scholar
  19. [19]
    Rota, G.C. Finite Operators Calculus. Academic Press: New York 1975.Google Scholar
  20. [20]
    Szegö, G. Orthogonal polynomials. American mathematical society Colloquium publ. vol.XXIII. Rew.ed New-York 1959.zbMATHGoogle Scholar
  21. [21]
    Sheffer, I.M. Note an Appell polynomials. Bull.Amer.Math. Soc. 51 (1945), 739–744.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    Shohat, J. The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer.J.Math. 58 (1936), 453–464.CrossRefMathSciNetGoogle Scholar
  23. [23]
    Simon, B. The P(4)2 Euclidean (Quantum) Field Theory. Princeton N.J.: Princeton University Press 1974.Google Scholar
  24. [24]
    Sun, T.C. On central and non-central limit theorems for non linear functions of a stationary Gaussian process. In: Dependence in Probability and Statistik.Google Scholar
  25. [25]
    Surgailis, D. Convergence of sums of non-linear functions of moving averages to self-similar processes. (Russian). Dokl.Akad.Nauk SSSR 257 (1981), No 1, 51–54.MathSciNetGoogle Scholar
  26. [26]
    Surgailis, D. Domains of attraction of self-similar multiple integrals. (Russian) Litovsk.Mat.Sb. 22 (1982), No 2, 195–201.MathSciNetGoogle Scholar
  27. [27]
    Surgailis, D. On Poisson multiple stochastic integrals and associated equillibrium Markov processes.–In: Lecture Notes Contr.Inform.Sci., 49, Springer: Berlin 1983, pp. 233–248.Google Scholar
  28. [28]
    Taqqu, M.S. Weak convergence to fractional Brownian motion and to Rosenblatt processes. Z.Wahrsch.verw.Gebibte 31 (1975), 287–302.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    Tagqu, M.S. Convergence of iterated processes of arbitrary Hermite rank. Z.Wanrsch.verw.Gebiete 50 (1979), 53–83.CrossRefGoogle Scholar
  30. [30]
    Yosida, K. Functional Analysis. Springer: Berlin-Gtitingen 1965.zbMATHGoogle Scholar

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© Springer Science+Business Media New York 1986

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  • L. Giraitis
  • D. Surgailis

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