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This paper is devoted to the estimation of the minimal dimension P of thestate-space realizations of a high-dimensional time series y, defined as anoisy version (the noise is white and Gaussian) of a useful signal with lowrank rational spectral density, in the high-dimensional asymptotic regime wherethe number of available samples N and the dimension of the time series Mconverge towards infinity at the same rate. In the classical low-dimensionalregime, P is estimated as the number of significant singular values of theempirical autocovariance matrix between the past and the future of y, or as thenumber of significant estimated canonical correlation coefficients between thepast and the future of y. Generalizing large random matrix methods developed inthe past to analyze classical spiked models, the behaviour of the abovesingular values and canonical correlation coefficients is studied in thehigh-dimensional regime. It is proved that they are smaller than certainthresholds depending on the statistics of the noise, except a finite number ofoutliers that are due to the useful signal. The number of singular values ofthe sample autocovariance matrix above the threshold is evaluated, is shown tobe almost independent from P in general, and cannot therefore be used toestimate P accurately. In contrast, the number s of canonical correlationcoefficients larger than the corresponding threshold is shown to be less thanor equal to P, and explicit conditions under which it is equal to P areprovided. Under the corresponding assumptions, s is thus a consistent estimateof P in the high-dimensional regime. The core of the paper is the developmentof the necessary large random matrix tools.

On the largest singular values of certain large random matrices with  application to the estimation of the minimal dimension of the state-space  representations of high-dimensional time series