Abstract
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.
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References
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)
Bramson, M., Griffeath, D.: Renormalizing the three-dimensional voter model. Ann. Probab. 7, 418–432 (1979)
Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)
Cameron, R.H., Martin, W.T.: The orthogonal development of nonlinear functionals in series of Fourier–Hermite functionals. Ann. Math. 48, 385–392 (1947)
Cramer, H.: On the theory of stationary random processes. Ann. Math. 41, 215–230 (1940)
Dawson, D., Ivanoff, G.: Branching diffusions and random measures. In: Advances in Probability. Dekker, New York (1979)
Dobrushin, R.L.: Gaussian and their subordinated generalized fields. Ann. Probab. 7, 1–28 (1979)
Dobrushin, R.L., Major, P.: Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 27–52 (1979)
Dobrushin, R.L., Major, P.L: On the asymptotic behaviour of some self-similar fields. Sel. Mat. Sov. 1(3), 293–302 (1981)
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, 515–533
Dobrushin, R.L., Minlos, R.A.: Polynomials of linear random functions. Uspekhi Mat. Nauk 32, 67–122 (1977)
Dynkin, E.B.: Die Grundlagen der Theorie der Markoffschen Prozesse, Band 108. Springer, Berlin (1961)
Eidlin, V.L., Linnik, Yu.V.: A remark on analytic transformation of normal vectors. Theory Probab. Appl. 13, 751–754 (1968) (in Russian)
Gelfand, I.M., Vilenkin, N.Ya.: Generalized Functions. IV. Some Applications of Harmonic Analysis. Academic (Harcourt, Brace Jovanovich Publishers), New York (1964)
Gross, L.: Logarithmic Soboliev inequalities. Am. J. Math. 97, 1061–1083 (1975)
Holley, R.A., Stroock, D.: Invariance principles for some infinite particle systems. In: Stochastic Analysis, pp. 153–173. Academic Press, New York (1978)
Itô, K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–164 (1951)
Kesten, H., Spitzer, F.: A limit theorem related to a class of self-similar processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 5–25 (1979)
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)
Lamperti, J.: Semi-stable stochastic processes. Trans. Am. Math. Soc. 104, 62–78 (1962)
Löwenstein, J.H., Zimmerman, W.: The power counting theorem for Feynman integrals with massless propagators. Commun. Math. Phys. 44, 73–86 (1975)
Major, P.: Renormalizing the voter model. Space and space-time renormalization. Stud. Sci. Math. Hung. 15, 321–341 (1980)
Major, P.: Limit theorems for non-linear functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 57, 129–158 (1981)
McKean, H.P.: Geometry of differential space. Ann. Probab. 1, 197–206 (1973)
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)
Nelson, E.: The free Markov field. J. Funct. Anal. 12, 211–227 (1973)
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)
Rosenblatt, M.: Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 125–132 (1979)
Rosenblatt, M.: Limit theorems for Fourier transform of functionals of Gaussian sequences. Z. Wahrscheinlichkeitstheorie verw. Gebiete 55, 123–132 (1981)
Segal, J.E.: Tensor algebras over Hilbert spaces. Trans. Am. Math. Soc. 81, 106–134 (1956)
Sinai, Ya.G.: Automodel probability distributions. Theory Probab. Appl. 21, 273–320 (1976)
Sinai, Ya.G.: Mathematical problems of the theory of phase transitions. Akadémiai Kiadó, Budapest with Pergamon Press (1982)
Taqqu, M.S.: Weak convergence of fractional Brownian Motion to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31, 287–302 (1975)
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)
Taqqu, M.S.: A representation for self-similar processes. Stoch. Process. Appl. 7, 55–64 (1978)
Taqqu, M.S.: Convergence of iterated process of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 53–83 (1979)
Totoki, H.: Ergodic Theory. Lecture Note Series, vol. 14. Aarhus University, Aarhus (1969)
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Major, P. (2014). History of the Problems: Comments. In: Multiple Wiener-Itô Integrals. Lecture Notes in Mathematics, vol 849. Springer, Cham. https://doi.org/10.1007/978-3-319-02642-8_9
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