History of the Problems: Comments

  • Péter Major
Part of the Lecture Notes in Mathematics book series (LNM, volume 849)


Here we summarize the content of the previous chapters. We explain the history of the results, give the necessary references to them, and also discuss the underlying motivations. We also present some results which are related to the subject of this work only in an indirect way, but they give a better insight into it.


Fractional Brownian Motion Generalize Field Hurst Parameter Gaussian Random Field Stationary Increment 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bleher, P.M., Missarov, M.D.: The equations of Wilson’s renormalization group and analytic renormalization. Commun. Math. Phys. 74(3), I. General results, 235–254, II. Solution of Wilson’s equations, 255–272 (1980)Google Scholar
  2. 2.
    Bramson, M., Griffeath, D.: Renormalizing the three-dimensional voter model. Ann. Probab. 7, 418–432 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear functionals in series of Fourier–Hermite functionals. Ann. Math. 48, 385–392 (1947)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cramer, H.: On the theory of stationary random processes. Ann. Math. 41, 215–230 (1940)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Dawson, D., Ivanoff, G.: Branching diffusions and random measures. In: Advances in Probability. Dekker, New York (1979)Google Scholar
  7. 7.
    Dobrushin, R.L.: Gaussian and their subordinated generalized fields. Ann. Probab. 7, 1–28 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 9.
    Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 27–52 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 10.
    Dobrushin, R.L., Major, P.L: On the asymptotic behaviour of some self-similar fields. Sel. Mat. Sov. 1(3), 293–302 (1981)MathSciNetGoogle Scholar
  10. 11.
    Dobrushin, R.L. Major, P., Takahashi, J.: Self-similar Gaussian fields. Finally it appeared as Major, P. (1982) On renormalizing Gaussian fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete 59, 515–533Google Scholar
  11. 12.
    Dobrushin, R.L., Minlos, R.A.: Polynomials of linear random functions. Uspekhi Mat. Nauk 32, 67–122 (1977)zbMATHGoogle Scholar
  12. 14.
    Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse, Band 108. Springer, Berlin (1961)CrossRefGoogle Scholar
  13. 15.
    Eidlin, V.L., Linnik, Yu.V.: A remark on analytic transformation of normal vectors. Theory Probab. Appl. 13, 751–754 (1968) (in Russian)MathSciNetGoogle Scholar
  14. 16.
    Gelfand, I.M., Vilenkin, N.Ya.: Generalized Functions. IV. Some Applications of Harmonic Analysis. Academic (Harcourt, Brace Jovanovich Publishers), New York (1964)Google Scholar
  15. 17.
    Gross, L.: Logarithmic Soboliev inequalities. Am. J. Math. 97, 1061–1083 (1975)Google Scholar
  16. 18.
    Holley, R.A., Stroock, D.: Invariance principles for some infinite particle systems. In: Stochastic Analysis, pp. 153–173. Academic Press, New York (1978)Google Scholar
  17. 19.
    Itô, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–164 (1951)CrossRefzbMATHGoogle Scholar
  18. 20.
    Kesten, H., Spitzer, F.: A limit theorem related to a class of self-similar processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 5–25 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 21.
    Kolmogorov, A.N.: Wienersche Spirale und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. Sci. U.R.S.S.(N.S.) 26, 115–118 (1940)Google Scholar
  20. 22.
    Lamperti, J.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104, 62–78 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 23.
    Löwenstein, J.H., Zimmerman, W.: The power counting theorem for Feynman integrals with massless propagators. Commun. Math. Phys. 44, 73–86 (1975)CrossRefGoogle Scholar
  22. 24.
    Major, P.: Renormalizing the voter model. Space and space-time renormalization. Stud. Sci. Math. Hung. 15, 321–341 (1980)zbMATHMathSciNetGoogle Scholar
  23. 25.
    Major, P.: Limit theorems for non-linear functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 57, 129–158 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 26.
    McKean, H.P.: Geometry of differential space. Ann. Probab. 1, 197–206 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 27.
    McKean, H.P.: Wiener’s theory of nonlinear noise (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1972). In: Stochastic Differential Equations, SIAM-AMS Proc., vol. VI, pp. 191–209. Amer. Math. Soc. Providence, RI (1973)Google Scholar
  26. 28.
    Nelson, E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973)CrossRefzbMATHGoogle Scholar
  27. 29.
    Rosenblatt, M.: Independence and dependence. In: Proceedings of Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 431–443. University of California Press, Berkeley (1962)Google Scholar
  28. 30.
    Rosenblatt, M.: Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 125–132 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 31.
    Rosenblatt, M.: Limit theorems for Fourier transform of functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 123–132 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 32.
    Segal, J.E.: Tensor algebras over Hilbert spaces. Trans. Am. Math. Soc. 81, 106–134 (1956)CrossRefzbMATHGoogle Scholar
  31. 33.
    Sinai, Ya.G.: Automodel probability distributions. Theory Probab. Appl. 21, 273–320 (1976)Google Scholar
  32. 34.
    Sinai, Ya.G.: Mathematical problems of the theory of phase transitions. Akadémiai Kiadó, Budapest with Pergamon Press (1982)Google Scholar
  33. 35.
    Taqqu, M.S.: Weak convergence of fractional Brownian Motion to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 287–302 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 36.
    Taqqu, M.S.: Law of the iterated logarithm for sums of non-linear functions of Gaussian variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 40, 203–238 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 37.
    Taqqu, M.S.: A representation for self-similar processes. Stoch. Process. Appl. 7, 55–64 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 38.
    Taqqu, M.S.: Convergence of iterated process of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 53–83 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 39.
    Totoki, H.: Ergodic Theory. Lecture Note Series, vol. 14. Aarhus University, Aarhus (1969)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Péter Major
    • 1
  1. 1.Alfréd Rényi Mathematical Institute Hungarian Academy of SciencesBudapestHungary

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