Abstract
We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.
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Ayache, A., Linde, W.: Approximation of Gaussian random fields: general results and optimal wavelet representation of the Lévy fractional motion. J. Theor. Probab., doi: 10.1007/s10959-007-0101-2 (2007)
Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Subsampling needlet coefficients on the sphere, arXiv:0706.4169v1 [math.ST] (2007)
Dzhaparidze, K., van Zanten, H.: A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004)
Dzhaparidze, K., van Zanten, H.: Optimality of an explicit series expansion of the fractional Brownian sheet. Stat. Probab. Lett. 71, 295–301 (2005)
Dzhaparidze, K., van Zanten, H., Zareba, P.: Representations of isotropic Gaussian random fields with homogeneous increments. J. Appl. Math. Stoch. Anal., doi: 10.1155/JAMSA/2006/72731 (2006). Article ID 72731, 25 pages
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. McGraw-Hill, New York (1953)
Iglói, E.: A rate-optimal trigonometric series expansion of the fractional Brownian motion. Electron. J. Probab. 10, 1381–1397 (2005)
Kolmogorov, A.: Wienerische Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C.R. (Doklady) Acad. Sci. USSR (N.S.) 26, 115–118 (1940)
Kühn, T., Linde, W.: Optimal series representation of fractional Brownian sheets. Bernoulli 8, 669–696 (2002)
Li, W.V., Linde, W.: Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27, 1556–1578 (1999)
Malyarenko, A.: Functional limit theorems for multiparameter fractional Brownian motion. J. Theor. Probab. 19, 263–288 (2006)
Mandelbrot, B., van Ness, J.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Vol. 1. Elementary Functions. Gordon & Breach, New York (1986)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Vol. 2. Special Functions, 2nd edn. Gordon & Breach, New York (1988)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Vol. 3. More Special Functions. Gordon & Breach, New York (1990)
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994)
Titchmatsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon, Oxford (1937)
Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A.: Probability Distributions on Banach Spaces. Reidel, Dordrecht (1988)
Van der Waart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York (1996)
Vilenkin, N.Y.: Special Functions and the Theory of Group Representations. American Mathematical Society, Providence (1968)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995)
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This work is supported in part by the Foundation for Knowledge and Competence Development and Sparbanksstiftelsen Nya.
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Malyarenko, A. An Optimal Series Expansion of the Multiparameter Fractional Brownian Motion. J Theor Probab 21, 459–475 (2008). https://doi.org/10.1007/s10959-007-0122-x
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DOI: https://doi.org/10.1007/s10959-007-0122-x