Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T12:25:49.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear estimation of self-similar processes via Lamperti's transformation

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Carl J. Nuzman  and
H. Vincent Poor
Carl J. Nuzman*
Affiliation:
Princeton University
H. Vincent Poor*
Affiliation:
Princeton University
*
∗Postal address: Dept. of Electrical Engineering, Engineering Quadrangle, Princeton, NJ 08544 5263, USA
∗Postal address: Dept. of Electrical Engineering, Engineering Quadrangle, Princeton, NJ 08544 5263, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

Keywords

Self-similar processesfractional Brownian motionlinear predictionwhiteninginnovationsscale-stationary processesLamperti's transformationWiener-Kolmogorov filter

MSC classification

Primary: 60G18: Self-similar processes
Secondary: 62M20: Prediction 60G25: Prediction theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 429 - 452
Copyright
Copyright © by the Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research supported in part by the US Office of Naval Research underGrant N00014–00–1–0141, and in part by the US Department of Defense NDSEG Fellowship Program.

References

Abramowitz, M. A., and Stegun, I. A. (1970). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Albin, J. M. P. (1998). On extremal theory for self-similar processes. Ann. Prob. 26, 743–793.Google Scholar
Barton, R. J., and Poor, H. V. (1988). Signal detection in fractional Gaussian noise. IEEE Trans. Inf. Theory 34, 943–959.Google Scholar
Burnecki, K., Maejima, M., and Weron, A. (1997). The Lamperti transformation for self-similar processes. Yokohama Math. J. 44, 25–42.Google Scholar
Coutin, L., and Decreusefond, L. (1999). Non-linear filtering theory in the presence of fractional Brownian motion. Ann. Appl. Prob. 9, 1058–1090.Google Scholar
Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Analysis 10, 177–214.Google Scholar
Erdérlyi, A. (ed.) (1954). Tables of Integral Transforms 1. McGraw-Hill, New York.Google Scholar
Gradshteyn, I. S., and Ryzhik, I. M. (1965). Table of Integrals, Series, and Products. Trans. ed. A. Jeffrey. Academic Press, New York.Google Scholar
Grigor'ev, S. V. (1965). Explicit extrapolation formulas for some probability processes. Kazan. Gos. Univ. Učen. Zap. 125, 106–109. [Engl. trans. in Selected Transl. in Math. Statist. and Prob. 11 (1973).]Google Scholar
Gripenberg, G., and Norros, I. (1996). On the prediction of fractional Brownian motion. J. Appl. Prob. 33, 400–410.CrossRefGoogle Scholar
Hurst, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 116, 770–808.Google Scholar
Jaglom, A. M. (1955). Correlation theory of processes with random stationary n-th increments. Matematika Sbornik 37, 141–196. [Engl. trans. in Amer. Math. Soc. Transl., Ser. 2, 8 (1958).]Google Scholar
Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78.Google Scholar
Leland, W. E., Taqqu, M. S., Willinger, W., and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Network 2, 1–14.Google Scholar
Loève, M. (1955). Probability Theory. Van Nostrand, New York.Google Scholar
Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman, San Francisco.Google Scholar
Mandelbrot, B. B., and Van Ness, J. (1968). Fractional Brownian motions, fractional noises, and applications. SIAM Rev. 10, 422–437.Google Scholar
Molchan, G. M. (1969). Gaussian processes with spectra which are asymptotically equivalent to a power of λ. Theor. Prob. Appl. 14, 530–532.Google Scholar
Molchan, G. M. and Golosov, Ju. I. (1969). Gaussian stationary processes with asymptotic power spectrum. Soviet Mathematics Doklady 10, 134–137.Google Scholar
Poor, H. V. (1994). An Introduction to Signal Detection and Estimation. Springer, New York.Google Scholar
Protter, M. H., and Morrey, C. B. (1977). A First Course in Real Analysis. Springer, New York.Google Scholar
Taqqu, M. S. (1985). A bibliographic guide to self-similar processes and long-range dependence. In Dependence in Probability and Statistics. Birkhäuser, Boston.Google Scholar
Vervaat, W. (1987). Properties of general self-similar processes. Bull. Int. Stat. Inst. 52, 199–216.Google Scholar
Willinger, W., Taqqu, M. S., and Teverovsky, V. (1999). Stock market prices and long-range dependence. Finance and Stochastics 3, 1–13.Google Scholar
Wong, E. (1971). Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.Google Scholar
Wornell, G. (1991). Synthesis, Analysis, and Processing of Fractal Signals. Ph.D. Thesis, MIT, Cambridge, MA.Google Scholar
Yazici, B., and Kashyap, R. L. (1997). A class of second-order stationary self-similar processes for 1/f phenomena. IEEE Trans. Signal Proc. 45, 396–410.Google Scholar