Application Of Large Random Matrices To Multivariate Time Series Analysis

A number of recent works proposed to use large random matrix theory in the context of high-dimensional statistical signal processing, traditionally modeled by a double asymptotic regime in which the dimension of the time series and the sample size both grow towards infinity. These contributions essentially addressed detection or estimation schemes depending on functionals of the sample covariance matrix of the observation. However, fundamental high-dimensional time series problems depend on matrices that are more complicated than the sample covariance matrix. The purpose of the present PhD is to study the behaviour of the singular values of 2 kinds of structured large random matrices, and to use the corresponding results to address an important statistical problem. More specifically, the observation (y_n)_{nin Z} is supposed to be a noisy version of a M-dimensional time series (u_n)_{nin Z} with rational spectrum that has some particular low rank structure, the additive noise (v_n)_{nin Z} being an independent identically distributed sequence of complex Gaussian vectors with unknown covariance matrix. An important statistical problem is the estimation of the minimal dimension P of the state space representations of u from N samples y_1,.., y_N. If L is any integer larger than P, the traditional approaches are based on the observation that P coincides with the rank of the autocovariance matrix R^L_{f|p} between the ML-dimensional random vectors (y_{n+L}^T,..,y_{n+2L-1}^T)^T and (y_{n}^T,.., y_{n+L-1}^T)^T, as well as with the number of non zero singular values of the normalized matrix C^L = (R^L)^{-1/2}R^L_{f|p} (R^L)^{-1/2} where R^L represents the covariance matrix of the above ML-dimensional vectors. In the low-dimensional regime where N->+infty while M and L are fixed, the matrices R^L_{f|p} and C^L can be consistently estimated by their empirical counterparts hat{R}^L_{f|p} and hat{C}^L, and P can be evaluated from the largest singular values of hat{R}^L_{f|p} and hat{C}^L. If however M and N->+infty in such a way that ML/N converges towards 0 c*