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Krein’s spectral theory and the Paley-Wiener expansion for fractional Brownian motion. (English) Zbl 1083.60028

Let \(\{X\},t\geq0\), be a fractional Brownian motion (fBm) with Hurst index \(H\in(0,1)\) and, for \(T>0\), let \(L_T\) be the closed Hilbert space spanned by the functions \(e_t(\lambda)=(e^{it\lambda}-1)/i\lambda\), \(t\in[0,T]\). The authors develop rich spectral theory for fBm. They represent the fBm as integral with respect to a fundamental martingale and obtain an isometry between \(L_T\) and the function space corresponding to the fundamental martingale. The isometry is a Fourier-type transformation whose kernel is given explcitly. The latter is used to turn \(L_T\) into a reproducing kernel Hilbert space. The authors then introduce functions whose role here is analogous to that of orthogonal polynomials in time series. This is used to obtain a closed form expression for the reproducing kernel and an orthonormal basis in \(L_T\).
The spectral theory developed by the authors allows them to generalise the classical Paley-Wiener expansion to the fractional case. They show that the complex fBm admits the series expansion \(\sum_{n} ((e^{2i\omega_nt}-1)/2i\omega_n)Z_n\), \(t\in[0,1],\) where the sum is over all integers, \(\omega_n\) are the real-valued zeroes of the Bessel function \(J_{1-H}\) and \(Z_n\) are independent complex Gaussian random variables. The classical result is indeed a particular case of this since for \(H=1/2\) the variables \(Z_n\) are standard Gaussian and \(\omega_n=n\pi\). The authors consider also the rate of convergence of the Paley-Wiener expansion of fBm and obtain an optimality property for it.

MSC:

60G15 Gaussian processes
60G35 Signal detection and filtering (aspects of stochastic processes)
60G51 Processes with independent increments; Lévy processes
62M15 Inference from stochastic processes and spectral analysis
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[1] Akhiezer, N. I. and Rybalko, A. M. (1968). Continual analogues of polynomials orthogonal on a circle. Ukrain. Mat. Zh. 20 3–24. · Zbl 0169.46601
[2] Ayache, A. and Taqqu, M. S. (2005). Approximating fractional Brownian motion by a random wavelet series: The rate optimality problem. J. Fourier Anal. Appl.
[3] Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214. · Zbl 0924.60034 · doi:10.1023/A:1008634027843
[4] Dzhaparidze, K. and Ferreira, J. A. (2002). A frequency domain approach to some results on fractional Brownian motion. Statist. Probab. Lett. 60 155–168. · Zbl 1014.60040 · doi:10.1016/S0167-7152(02)00307-3
[5] Dzhaparidze, K. and Van Zanten, J. H. (2004). A series expansion of fractional Brownian motion. Probab. Theory Related Fields 130 39–55. · Zbl 1059.60048 · doi:10.1007/s00440-003-0310-2
[6] Dzhaparidze, K. and Van Zanten, J. H. (2005). Optimality of an explicit series expansion of the fractional Brownian sheet. Statist. Probab. Lett. · Zbl 1099.60024
[7] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Higher Transcendental Functions II . McGraw-Hill, New York. · Zbl 0052.29502
[8] Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications . Univ. California Press, Berkeley. · Zbl 0080.09501
[9] Gripenberg, G. and Norros, I. (1996). On the prediction of fractional Brownian motion. J. Appl. Probab. 33 400–410. · Zbl 0861.60049 · doi:10.2307/3215063
[10] Hochstadt, H. (1971). The Functions of Mathematical Physics . Wiley, New York. · Zbl 0217.39501
[11] Hult, H. (2003). Approximating some Volterra type stochastic integrals with applications to parameter estimation. Stochastic Process. Appl. 105 1–32. · Zbl 1075.60532 · doi:10.1016/S0304-4149(02)00250-8
[12] Kailath, T., Vieira, A. and Morf, M. (1978). Inverses of Toeplitz operators, innovations, and orthogonal polynomials. SIAM Rev. 20 106–119. · Zbl 0382.47013 · doi:10.1137/1020006
[13] Krein, M. G. (1955). Continuous analogues of propositions on polynomials orthogonal on the unit circle. Dokl. Akad. Nauk SSSR 105 637–640.
[14] Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. Bernoulli 8 669–696. · Zbl 1012.60074
[15] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces . Springer, Berlin. · Zbl 0748.60004
[16] Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien . Deuxième édition revue et augmentée. Gauthier-Villars & Cie, Paris. · Zbl 0137.11602
[17] Li, W. V. and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27 1556–1578. · Zbl 0983.60026 · doi:10.1214/aop/1022677459
[18] Li, W. V. and Shao, Q. M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In Stochastic Processes : Theory and Methods (C. R. Rao and D. Shanbhag, eds.) 533–597. North-Holland, Amsterdam. · Zbl 0987.60053
[19] Molchan, G. M. (2003). Linear problems for a fractional Brownian motion: Group approach. Theory Probab. Appl. 47 69–78. · Zbl 1035.60084 · doi:10.1137/S0040585X97979445
[20] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 571–587. · Zbl 0955.60034 · doi:10.2307/3318691
[21] Nuzman, C. J. and Poor, H. V. (2000). Linear estimation of self-similar processes via Lamperti’s transformation. J. Appl. Probab. 37 429–452. · Zbl 0963.60034 · doi:10.1239/jap/1014842548
[22] Paley, R. E. and Wiener, N. (1934). Fourier Transforms in the Complex Domain VIII . Amer. Math. Soc., New York. · Zbl 0011.01601
[23] Pipiras, V. and Taqqu, M. S. (2001). Are classes of deterministic integrands for fractional Brownian motion on an interval complete? Bernoulli 7 873–897. · Zbl 1003.60055 · doi:10.2307/3318624
[24] Rudin, W. (1987). Real and Complex Analysis , 3rd ed. McGraw-Hill, New York. · Zbl 0925.00005
[25] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives . Gordon and Breach Science Publishers, Yverdon. · Zbl 0818.26003
[26] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes . Chapman and Hall, New York. · Zbl 0925.60027
[27] Watson, G. N. (1944). A Treatise on the Theory of Bessel Functions . Cambridge Univ. Press. · Zbl 0063.08184
[28] Yaglom, A. M. (1987). Correlation Theory of Stationary and Related Random Functions I . Springer, New York. · Zbl 0685.62078
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