Fractional Brownian Motions, Fractional Noises and Applications

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Previous article Next article Fractional Brownian Motions, Fractional Noises and ApplicationsBenoit B. Mandelbrot and John W. Van NessBenoit B. Mandelbrot and John W. Van Nesshttps://doi.org/10.1137/1010093PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. Adelman, Long cycles—fact or artifact?, Amer. Economic Rev., 60 (1965), 444–463 Google Scholar[2] William Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statistics, 22 (1951), 427–432 MR0042626 0043.34201 CrossrefISIGoogle Scholar[3] I. M. Gel'fand and , N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]xiv+384 MR0435834 Google Scholar[4] C. W. J. Granger, The typical spectral shape of an economic variable, Econometrica, 34 (1966), 150–161 CrossrefISIGoogle Scholar[5] G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. Soc., 71 (1951), 38–69 MR0051340 0043.30601 CrossrefISIGoogle Scholar[6] H. E. Hurst, , R. P. Black and , Y. M. Sinaika, Long Term Storage in Reservoirs. An Experimental Study, Constable, London, 1965 Google Scholar[7] A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), 115–118 MR0003441 0022.36001 Google Scholar[8] John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc., 104 (1962), 62–78 MR0138128 0286.60017 CrossrefGoogle Scholar[9] Paul Lévy, Random functions: General theory with special reference to Laplacian random functions, Univ. California Publ. Statist., 1 (1953), 331–390 MR0055607 0052.14402 Google Scholar[10] Michel Loève, Probability theory, 2nd ed. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-New York-London, 1960xvi+685 MR0123342 0095.12201 Google Scholar[11] Benoı⁁t Mandelbrot, Une classe processus stochastiques homothétiques à soi; application à la loi climatologique H. E. Hurst, C. R. Acad. Sci. Paris, 260 (1965), 3274–3277 MR0176521 0127.09501 Google Scholar[12] B. Mandelbrot, Self-similar error-clusters in communication systems and the concept of conditional stationarity, IEEE Trans. Comet. Tech., COM-13 (1965), 71–90 10.1109/TCOM.1965.1089090 CrossrefISIGoogle Scholar[13] B. Mandelbrot, Noises with an $l/f$ spectrum, a bridge between direct current and white noise, IEEE Trans. Information Theory, IT-13 (1967), 289–298 10.1109/TIT.1967.1053992 0148.40507 CrossrefISIGoogle Scholar[14] Benoit Mandelbrot, Sporadic random functions and conditional spectral analysis: Self-similar examples and limitsProc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. III: Physical Sciences, Univ. California Press, Berkeley, Calif., 1967, 155–179 MR0224243 0189.18302 Google Scholar[15] B. Mandelbrot and , J. R. Wallis, Noah, Joseph and operational hydrology, Water Resources Research, to appear Google Scholar[16] B. Mandelbrot and , J. R. Wallis, Computer experiments with fractional Gaussian noise, Water Resources Research, to appear Google Scholar[17] B. Mandelbrot and , J. R. Wallis, Some long run properties of geophysical records, Water Resources Research, to appear. Google Scholar[18] G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyusyu Univ. A., 4 (1949), 45–106 MR0032127 0045.40602 CrossrefGoogle Scholar[19] M. Rosenblatt, Independence and dependenceProc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, 431–443 MR0133863 0105.11802 Google Scholar[20] G. I. Taylor, Statistical theory of turbulence, Proc. Roy. Sac. Ser. A, 151 (1935), 421–478 CrossrefGoogle Scholar[21] H. Weyl, Bemerkungen zum Begriff der Differential-Quotenten gebrochener Ordnung, Vierteljschr. Naturforsch. Ges. Zürich, 62 (1967), 296–302 Google Scholar[22] A. M. Yaglom, Correlation theory of processes with random stationary nth increments, Amer. Math. Soc. Transl. (2), 8 (1958), 87–141 MR0091554 0080.34903 CrossrefGoogle Scholar[23] A. M. Yaglom, L. M. LeCam and , J. Neyman, Stationary Gaussian processes satisfying the strong mixing condition and best predictable functionalsBernoulli-Bayes-Laplace Anniversary Volume, Springer-Verlag, New York, 1965, 241–252 0142.13904 CrossrefGoogle Scholar[24] A. M. Yaglom, Outline of some topics in linear extrapolation of stationary random processesProc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II:Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif., 1967, 259–278 MR0215358 0189.18401 Google Scholar[25] Antoni Zygmund, Trigonometrical series, Chelsea Publishing Co., New York, 1952vi+329, 2nd ed. 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