https://hal-imt.archives-ouvertes.fr/hal-00604392Hachem, WalidWalidHachemLTCI - Laboratoire Traitement et Communication de l'Information - Télécom ParisTech - IMT - Institut Mines-Télécom [Paris] - CNRS - Centre National de la Recherche ScientifiqueMoulines, E.E.MoulinesLTCI - Laboratoire Traitement et Communication de l'Information - Télécom ParisTech - IMT - Institut Mines-Télécom [Paris] - CNRS - Centre National de la Recherche ScientifiqueRoueff, FrançoisFrançoisRoueffLTCI - Laboratoire Traitement et Communication de l'Information - Télécom ParisTech - IMT - Institut Mines-Télécom [Paris] - CNRS - Centre National de la Recherche ScientifiqueError Exponents for Neyman-Pearson Detection of a Continuous-Time Gaussian Markov Process From Regular or Irregular SamplesHAL CCSD2011Kalman filterNeyman-Pearson detectionStein's Lemmastochastic differential equationsstochastic differential equations.Error exponentsGaussian Markov processes[INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT][MATH.MATH-IT] Mathematics [math]/Information Theory [math.IT]Télécom Paristech, Admin2011-06-28 19:08:122020-10-19 12:07:092011-06-28 19:08:13enJournal articles10.1109/TIT.2011.21336301This paper addresses the detection of a stochastic process in noise from a finite sample under various sampling schemes. We consider two hypotheses. The noise only hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The signal plus noise hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants and corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by error exponents that are completely characterized. It turns out that they are related to the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks.