In this talk, we consider the sample correlation matrix $R$ associated to $n$ observations of a $p$-dimensional time series. In our framework, we allow that $p/n$ may tend to 0 or a positive constant. If the time series has a finite fourth moment, we show that the sample correlation matrix can be approximated by its sample covariance counterpart for a wide variety of models.

This result is very important for data analysts who use principal component analysis to detect some structure in high-dimensional time series. From a theoretical point of view, it allows to derive a plethora of ancillary results for functionals of the eigenvalues of $R$. For instance, we determine the almost sure behavior of the largest and smallest eigenvalues, and the limiting spectral distribution of $R$.

Finally, we discuss the case of time series with infinite fourth moment and determine the optimal moment conditions for the convergence of the empirical spectral distributions to their usual limits.