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Behaviour of non parametric estimators of second order statistics of high dimensional time series : a large random matrix approach

Abstract : Large random matrices have been proved to be of fundamental importance in mathematics (high dimensional probability, operator algebras, combinatorics, number theory,...) and in physics (nuclear physics, quantum fields theory, quantum chaos,..) for a long time. The use of large random matrices is more recent in statistical signal processing and time series analysis. The corresponding tools turn out to be useful when the observation is a large dimension (say M) multivariate time series and the sample size N is not much larger than M, a situation that becomes very common due to the spectacular development of data acquisition devices and sensor networks. This context poses several new difficult statistical problems that are intensively studied by the high-dimensional statistics community. The most significant example is related to the fundamental problem of estimating the covariance matrix of the observation because the standard empirical covariance matrix is known to perform poorly if N is not significantly larger than M. As a result, the conventional statistical inference schemes that are based on functionals of the empirical covariance matrix may perform poorly. To mitigate this conceptual difficulty, the most popular approaches were based on the design of inference schemes using some possible degree of sparsity of the underlying parameters. However, sparsity is a property that does not necessarily hold. The use of large random matrix theory is an appealing alternative because, under some assumptions on the observations, it is possible to precise the behaviour of certain functionals of the empirical covariance matrix when M and N are both large, and to use the corresponding results to design new improved performance inference schemes. While some papers produced several valuable results, considerable work remains to be done to exploit the potential of large random matrix technics in the context of statistics of high-dimensional Gaussian time series. The proposed works in this manuscript are thus at the interface between large random matrices and the statistics of multivariate time series
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Submitted on : Wednesday, May 11, 2022 - 2:14:11 PM
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Alexis Rosuel. Behaviour of non parametric estimators of second order statistics of high dimensional time series : a large random matrix approach. Statistics [math.ST]. Université Gustave Eiffel, 2021. English. ⟨NNT : 2021UEFL2034⟩. ⟨tel-03665080⟩

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