# Applications of large randommatrix to high dimensional statistical signal processing

Abstract : This thesis focuses on statistical problems involving a multivariate time series ${\bf y}_n$ of large dimension $M$ defined as the sum of gaussian white noise temporally and spatially and a useful signal defined as the output of an unknown finite impulse response single input multiple outputs system driven by a deterministic scalar nonobservable sequence. Supposing $({\bf y}_n)_{n=1,...,N}$ is available, a number of existing methods are based on the functionals of empirical covariance matrix $\hat{\bf R}_L$ of $ML$--dimensional vectors $({\bf y}^{(L)}_n)_{n=1,...,N}$ obtained by stacking the vectors $({\bf y}_k)_{k=n,...,n+L-1}$, where $L$ is a relevant parameter. In the case where the number of observations $N$ is much larger than $ML$ the dimension of vectors $({\bf y}^{(L)}_n)_{n=1,...,N}$, $\hat{\bf R}_L$ behaves as its mathematical expectation in the sense of spectral norm. This allows us to study the inference technique based on $\hat{\bf R}_L$ via classical techniques of asymptotic statistics. In this thesis, we interested in the case where $ML$ and $N$ have the same order of magnitude, we call this the asymptotic regimes in which $M$ and $N$ converge towards infinity, such that the ratio $\frac{ML}{N}$ converges towards a strictly positive constant, given that $L$ may scale with $M,N$. To solve the problems in this work, it is necessary to investigate the behaviour of the eigenvalues and eigenvectors of the random matrix $\hat{\bf R}_L$. Taking account of the particular structure of vectors $({\bf y}^{(L)}_n)_{n=1,...,N}$, $\hat{\bf R}_L$ coincides with the Gram matrix of a block-Hankel matrix ${\bf \Sigma }_L$, and this specificity requires the development of appropriate techniques. We interested to the case where the number of coefficients $P$ of the finite impulse response generated the useful signal and the parameter $L$ remain fixed when $M,N$ grow large. As a consequence, the matrix ${\bf \Sigma }_L$ is a finite rank perturbation of block-Hankel matrix ${\bf W}_L$ composed of additive noise. We prove that eigenvalues and eigenvectors of $\hat{\bf R}_L$ behave as if the entries of matrix ${\bf W}_L$ are independent and identically distributed. This allows us to construct detection tests of useful signal based on largest eigenvalues of $\hat{\bf R}_L$ and to develop new estimation strategies of the regularization parameter of the spatio-temporal Wiener filter estimated from a training sequence. This approach is characterized by the asymptotic behaviour of the resolvent of matrix $\hat{\bf R}_L$. We also prove that these results provide consistent subspace estimation methods for source localization using spatial-smoothing scheme. Motivated by the case where $P$ and $L$ may converge towards infinity, we move off somewhat the initial model. We suppose that the matrix ${\bf \Sigma }_L$ is the sum of the block-Hankel random matrix ${\bf W}_L$ with a deterministic matrix without particular structure. Using the approaches based on Stieltjes transform and tools adapted to gaussian noise, we prove that the empirical eigenvalue distribution of $\hat{\bf R}_L$ has deterministic behaviour which we shall describe. Provided $\frac{L^2}{MN}$ converges towards $0$, we do likewise for the elements of the resolvent of $\hat{\bf R}_L$. Finally, we return to the initial model, but we suppose that $P,L$ converge towards infinity with the same rate. In this context, matrix ${\bf \Sigma }_L$ is a matrix whose rank goes to infinity, and thus the techniques employed in chapter 2 are not applicable. Using the results obtained in chapter 3, we establish that when $\frac{L^2}{MN}$ goes to $0$, the elements of the resolvent of $\hat{\bf R}_L$ behave as the elements of a deterministic matrix which coincides with the deterministic equivalent of the resolvent of the information plus noise model in which entries of the noise matrix are independent and identically distributed. In the case where $\frac{L}{M}$ goes to $0$, this allows us to extend the results of chapter 2 related to the determination of the regularization parameter of the spatial-temporal Wiener filter estimated from a training sequence.
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Gia-Thuy Pham. Applications of large randommatrix to high dimensional statistical signal processing. Statistics [math.ST]. Université Paris Est Marne la Vallée, 2017. English. ⟨tel-01712290⟩

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