Abstract.
Let B be a fractional Brownian motion with Hurst index H∈(0,1). Denote by the positive, real zeros of the Bessel function J −H of the first kind of order −H, and let be the positive zeros of J 1−H . In this paper we prove the series representation where X 1 ,X 2 ,... and Y 1 ,Y 2 ,... are independent, Gaussian random variables with mean zero and and the constant c H 2 is defined by c H 2=π−1Γ(1+2H) sin πH. We show that with probability 1, both random series converge absolutely and uniformly in t∈[0,1], and we investigate the rate of convergence.
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Ayache, A., Taqqu, M.S.: Approximating fractional Brownian motion by a random wavelet series: the rate optimality problem. J. Fourier Anal. Appl. To appear 2003
Dzhaparidze, K., Van~Zanten, J.H.: Optimality of an explicit series expansion of the fractional Brownian sheet. Preprint, 2003
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.:Tables of integral transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954a
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.:Tables of integral transforms. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954b
Hochstadt, H.: The functions of mathematical physics. Wiley-Interscience, New York- London-Sydney, 1971
Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum. C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 26, 115–118 (1940)
Kühn, T., Linde, W.: Optimal series representation of fractional Brownian sheets. Bernoulli 8 (5), 669–696 (2002)
Macaulay-Owen, P.: Parseval’s theorem for Hankel transforms. Proc. Lond. Math. Soc., II. Ser. 45, 458–474 (1939)
Mandelbrot, B.B., Van~Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
Meyer, Y., Sellan, F., Taqqu, M.S.: Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. J. Fourier Anal. Appl. 5 (5), 465–494 (1999)
Paley, R.E.A.C., Wiener, N.:Fourier transforms in the complex domain. AMS, New York, 1934
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon, 1993
Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random processes. Chapman & Hall, New York, 1994
Titchmarsh, E.C.: Introduction to the theory of Fourier integrals. Clarendon Press, Oxford, 1937
Van~der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes with applications to statistics. Springer-Verlag, New York, 1996
Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press, Cambridge, England, 1944
Yaglom, A.M.: Correlation theory of stationary and related random functions. Vol. I. Springer-Verlag, New York, 1987
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Mathematics Subject Classification (2000): 60G15, 60G18, 33C10
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Dzhaparidze, K., Zanten, H. A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004). https://doi.org/10.1007/s00440-003-0310-2
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DOI: https://doi.org/10.1007/s00440-003-0310-2