Summary
Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been n α, with 1/2<α<1 where n is the sample size. Here examples of limit theorems are given for quadratic functions with long range memory (not instantaneous) with a normalization n α, 0<α<1/2.
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Dobrushin, R.L.: Gaussian and their subordinated self-similar random generalized fields. Ann. Probability 7, 1–28 (1979)
Ibragimov, I.A., Linnik, Yu.V.: Independent and Stationary Sequences of Random Variables. Gröningen: Walters-Noordhoff 1971
Rosenblatt, M.: Independence and dependence. Proc. 4th Berkeley Sympos. Math. Statist. Probab., 431–443. Univ. Calif. (1961)
Rosenblatt, M.: Fractional integrals of stationary processes and the central limit theorem. J. Appl. Probability 13, 723–732 (1976)
Sun, T.C.: Some further results on central limit theorems for non-linear functions of a normal stationary process. J. Math. Mech., 14, 71–85 (1965)
Taqqu, M.S.: Weak convergence to fractional Brownian motion and 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 that exhibit a long range dependence. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 40, 203–238 (1977)
Taqqu, M.S.: A representation for self-similar processes, manuscript
Zygmund, A.: Trigonometric Series. Cambridge: Cambridge University Press 1968
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Research supported in part by the Office of Naval Research
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Rosenblatt, M. Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables. Z. Wahrscheinlichkeitstheorie verw Gebiete 49, 125–132 (1979). https://doi.org/10.1007/BF00534252
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DOI: https://doi.org/10.1007/BF00534252