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Power spectra of random spike fields and related processes

Part of: Communication, information Stochastic processes Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pierre Brémaud ,
Laurent Massoulié  and
Andrea Ridolfi
Pierre Brémaud*
Affiliation:
École Polytechnique Fédérale de Lausanne and INRIA-ENS
Laurent Massoulié*
Affiliation:
Microsoft Research
Andrea Ridolfi*
Affiliation:
École Polytechnique Fédérale de Lausanne
*
∗ Postal address: INRIA-ENS, Département d' Informatique, École Normale Supérieure, 45 rue d'Ulm, F-75005 Paris, France. Email address: [email protected]
∗∗ Postal address: Microsoft Research, 7 J. J. Thomson Avenue, Cambridge CB3 0FB, UK.
∗∗∗ Postal address: School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland.
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Abstract

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In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

Keywords

Shot noiserandom samplingpoint processBochner power spectral measureBartlett power spectral measureHawkes processrandom fieldbranching processlinear birth-death process

MSC classification

Primary: 60G12: General second-order processes 60G35: Signal detection and filtering 62M15: Spectral analysis 62M40: Random fields; image analysis 60G55: Point processes 60G60: Random fields 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G20: Generalized stochastic processes 62M30: Spatial processes 94A05: Communication theory 94A20: Sampling theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1116 - 1146
Copyright
Copyright © Applied Probability Trust 2005 

References

Abry, P. and Flandrin, P. (1996). Point processes, long-range dependence and wavelets. In Wavelets in Medicine and Biology, eds Aldroubi, A. and Unser, M., CRC Press, Boca Raton, FL, pp. 413–437.Google Scholar
Abry, P. and Veitch, D. (1998). Wavelet analysis of long-range-dependent traffic. Trans. Inf. Theory 44, 2–15.CrossRefGoogle Scholar
Baccelli, F. and Blaszczyszyn, B. (2001). On a coverage process ranging from the Boolean model to the Poisson-Voronoi tessellation with applications to wireless communications. Adv. Appl. Prob. 33, 293–323.CrossRefGoogle Scholar
Beutler, F. J. and Leneman, O. A. Z. (1966). Random sampling of random processes. I. Stationary point processes. Inf. Control 9, 325–346.CrossRefGoogle Scholar
Beutler, F. J. and Leneman, O. A. Z. (1968). The spectral analysis of impulse processes. Inf. Control 12, 236–258.CrossRefGoogle Scholar
Bondesson, L. (1988). Shot-noise processes and shot-noise distributions. In Encyclopedia of Statistical Sciences, Vol. 8, John Wiley, New York, pp. 448–452.Google Scholar
Brémaud, P. (2002). Mathematical Principles of Signal Processing. Fourier and Wavelet Analysis. Springer, New York.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (2001). Hawkes branching processes without ancestors. J. Appl. Prob. 38, 122–135.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (2002). Power spectra of general shot noises and Hawkes point processes with a random excitation. Adv. Appl. Prob. 34, 205–222.CrossRefGoogle Scholar
Brillinger, D. R. (1972). The spectral analysis of stationary interval functions. In Proc. Sixth Berkeley Symp. Math. Statist. Prob., Vol. 1, University of California Press, Berkeley, CA, pp. 483–513.Google Scholar
Brillinger, D. R. (1981). Time Series. Data Analysis and Theory, 2nd edn. Holden-Day, Oakland, CA.Google Scholar
Daley, D. J. (1970). Weakly stationary point processes and random measures. J. R. Statist. Soc. B 33, 406–428.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2002). An Introduction to the Theory of Point Processes, Vol. 1, Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.CrossRefGoogle Scholar
Hawkes, A. G. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.CrossRefGoogle Scholar
Klüppelberg, C. and Mikosch, T. (1995). Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli 1, 125–147.CrossRefGoogle Scholar
Leneman, O. A. Z. (1966). Random sampling of random processes. II. Impulse processes. Inf. Control 9, 347–363.CrossRefGoogle Scholar
Leneman, O. A. Z. and Lewis, J. B. (1966). Random sampling of random processes. Mean-square comparison of various interpolators. Trans. Automatic Control 11, 396–403.CrossRefGoogle Scholar
Masry, E. (1978). Alias-free sampling. An alternative conceptualization and its applications. Trans. Inf. Theory 24, 317–324.CrossRefGoogle Scholar
Masry, E. (1978). Poisson sampling and spectral estimation of continuous-time processes. Trans. Inf. Theory 24, 173–183.CrossRefGoogle Scholar
Moore, M. and Thompson, P. (1991). Impact of Jittered sampling on conventional spectral estimates. J. Geophys. Res. 96, 18519–18526.CrossRefGoogle Scholar
Neveu, J. (1977). Processus ponctuels. In École d'Été de Probabilités de Saint-Flour VI (Lecture Notes Math. 598), Springer, Berlin, pp. 249–445.Google Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrence and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.CrossRefGoogle Scholar
Ridolfi, A. (2004). Power spectra of random spikes and related complex signals, with application to communications. . École Polytechnique Fédérale de Lausanne. Available at http://infoscience. epfl.ch/search.py?recid=33626&ln=en.Google Scholar
Ridolfi, A. and Win, M. Z. (2005). Power spectra of multipath faded pulse trains. In Proc. Internat. Symp. Inf. Theory, IEEE, pp. 102–106.Google Scholar
Ridolfi, A. and Win, M. Z. (2005). Ultrawide bandwidth signals as shot-noise: a unifying approach. To appear in J. Select. Areas Commun. (special issue on Ultra-Wideband Wireless Communications: Theory and Applications).Google Scholar
Samorodnitsky, G. (1995). A class of shot noise models for financial applications. In Athens Conf. Appl. Prob. Time Series Anal. (Lecture Notes Statist. 114), Vol. 1, eds Heyde, C. C. et al., Springer, New York, pp. 332–353.Google Scholar
Schottky, W. (1918). Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern. Ann. Physik 57, 541–567.CrossRefGoogle Scholar
Shapiro, H. S. and Silverman, R. A. (1960). Alias-free sampling of random noise. J. Soc. Indust. Appl. Math. 8, 225–248.CrossRefGoogle Scholar
Vere-Jones, D. and Davies, R. B. (1966). A statistical survey of earthquakes in the main seismic area of New Zealand. Part II: Time series analysis. N. Z. J. Geol. Geophys. 9, 251–284.Google Scholar