Given $n$ i.i.d. samples $(\boldsymbol{\vec x}_1, \cdots, \boldsymbol{\vec x}_n)$ of a $N$-dimensional long memory stationary process, it has recently been proved that the limiting spectral distribution of the sample covariance matrix,
$$
\frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i
$$
has an unbounded support as $N,n\to \infty$ and $\frac Nn\to c\in (0,\infty)$. As a consequence, its largest eigenvalue
$$
\lambda_{\max} \left( \frac 1n \sum_{i=1}^n \boldsymbol{\vec x}_i \boldsymbol{\vec x}^*_i
\right)
$$
goes to infinity. In this talk, we will describe its asymptotics and fluctuations, tightly related to the features of the underlying population covariance matrix, which is of a Toeplitz nature.


This is a joint work with Peng Tian and Florence Merlevède.