Summary
In a previous paper the authors obtained a functional law of the iterated logarithm for a class of self-similar processes\(\bar X\) with stationary increments, which are represented by multiple Wiener integrals. This result is extended to a certain class of processes represented by multiple Wiener integrals which converge to\(\bar X\) with an appropriate normalization. As an application a functional log log law for nonlinear functionals of some stationary Gaussian processes is given.
References
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
de Haan, L.: On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts: Amsterdam 1970
Dobrushin, R.L., Major, P.: Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheor. Verw. Geb.50, 27–52 (1979)
Kôno, N.: Classical limit theorems for dependent random sequences having moment conditions. Probability Theory and Mathematical Statistics, Lecture Notes in Math.1021, 315–319, Berlin, Heidelberg, New York: Springer 1983
Lai, T.L., Stout, W.: Limit theorems for sums of dependent random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.51, 1–14 (1980)
Mori, T., Oodaira, H.: The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals. Probab. Th. Rel. Fields71, 367–391 (1986)
Plikusas, A.: Some properties of the multiple Ito integral (in Russian). Liet. Mat. Rink.21, 163–173 (1981)
Taqqu, M.S.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheor. Verw. Geb.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. Wahrscheinlichkeitstheor. Verw. Geb.40, 203–218 (1977)
Taqqu, M.S.: Convergence to integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheor. Ver. Geb.50, 53–83 (1979)
Taqqu, M.S.: Self-similar processes and related ultraviolet and infrared catastrophes. Colloquia Mathematics Societatis Janos Bolyai27, Random Fields. Esztergom, Hungary 1979
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Mori, T., Oodaira, H. The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals. Probab. Th. Rel. Fields 76, 299–310 (1987). https://doi.org/10.1007/BF01297487
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DOI: https://doi.org/10.1007/BF01297487
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Gaussian Process
- Iterate Logarithm