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Power spectra of random spike fields and related processes

Part of: Communication, information Markov processes Inference from stochastic processes 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: pierre.bremaud@ens.fr
∗∗ 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 

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