  This paper is devoted to the estimation of the minimal dimension P of the
state-space realizations of a high-dimensional time series y, defined as a
noisy version (the noise is white and Gaussian) of a useful signal with low
rank rational spectral density, in the high-dimensional asymptotic regime where
the number of available samples N and the dimension of the time series M
converge towards infinity at the same rate. In the classical low-dimensional
regime, P is estimated as the number of significant singular values of the
empirical autocovariance matrix between the past and the future of y, or as the
number of significant estimated canonical correlation coefficients between the
past and the future of y. Generalizing large random matrix methods developed in
the past to analyze classical spiked models, the behaviour of the above
singular values and canonical correlation coefficients is studied in the
high-dimensional regime. It is proved that they are smaller than certain
thresholds depending on the statistics of the noise, except a finite number of
outliers that are due to the useful signal. The number of singular values of
the sample autocovariance matrix above the threshold is evaluated, is shown to
be almost independent from P in general, and cannot therefore be used to
estimate P accurately. In contrast, the number s of canonical correlation
coefficients larger than the corresponding threshold is shown to be less than
or equal to P, and explicit conditions under which it is equal to P are
provided. Under the corresponding assumptions, s is thus a consistent estimate
of P in the high-dimensional regime. The core of the paper is the development
of the necessary large random matrix tools.
