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Weak convergence to fractional Brownian motion and to the Rosenblatt process. (English) Zbl 0303.60033

60G10Stationary processes
60G15Gaussian processes
60J65Brownian motion
60F05Central limit and other weak theorems
Full Text: DOI
[1] Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968 · Zbl 0172.21201
[2] Davydov, Y.A.: The invariance principle for stationary processes. Theor. Probability Appl. 15, 487-498 (1970) · Zbl 0219.60030 · doi:10.1137/1115050
[3] de Haan, L.: On regular variation and its application to the weak convergence of sample extremes. Math. Centre Tracts 32, Math. Centre, Amsterdam (1970) · Zbl 0226.60039
[4] Feller, W.: An Introduction to Probability Theory and its Applications 2. 2nd ed. New York: Wiley 1971 · Zbl 0219.60003
[5] Gisselquist, R.: A continuum of collision process limit theorems. Ann. Probability 1, 231-239 (1973) · Zbl 0263.60047 · doi:10.1214/aop/1176996976
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[7] Lukacs, E.: Characteristic Functions. 2nd ed. New York: Hafner 1970 · Zbl 0201.20404
[8] Mandelbrot, B.: Limit theorems on the self-normalized range for weakly and strongly dependent processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 31, 271-285 (1975) · Zbl 0288.60033 · doi:10.1007/BF00532867
[9] Mandelbrot, B.: Statistical methodology for non-periodic cycles: from the covariance to R/S analysis. Ann. Econ. and Social Measurement 1, 259-290 (1972)
[10] Mandelbrot, B., McCamy, K.: On the secular pole motion and the Chandler wobble. Geophys. J. Roy. Astron. Soc. 21, 217-232 (1970)
[11] Mandelbrot, B., van Ness, J.W.: Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422-437 (1968) · Zbl 0179.47801 · doi:10.1137/1010093
[12] Mandelbrot, B., Wallis, J.R.: Noah, Joseph and Operational hydrology. Water Resources Research 4, 909-918 (1968) · doi:10.1029/WR004i005p00909
[13] Mandelbrot, B., Wallis, J.R.: Robustness of the rescaled range and the measurement of the long run statistical dependence. Water Resources Research 5, 967-988 (1969) · doi:10.1029/WR005i005p00967
[14] Rosenblatt, M.: Independence and dependence. Proc. 4th Berkeley Sympos. Math. Statist. Probab. pp. 411-443. Berkeley: Univ. Calif. Press 1961 · Zbl 0105.11802
[15] Rozanov, Y.A.: Stationary Random Processes. San Francisco: Holden-Day 1967 · Zbl 0152.16302
[16] Spitzer, F.: Uniform motion with elastic collision of an infinite particle system. J. Math. Mech. 18, 973-989 (1969) · Zbl 0184.21102
[17] 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) · Zbl 0138.10803
[18] Taqqu, M.: Note on evaluations of R/S for fractional noises and geophysical records. Water Resources Research 6, 349-350 (1970) · doi:10.1029/WR006i001p00349
[19] Taqqu, M.: Limit theorems for sums of strongly dependent random variables. Ph.D. thesis, Columbia Univ. (1972)