Deformation of Marchenko-Pastur distribution for the correlated time series

We study the eigenvalue of the Wishart matrix, which is created from a time series with temporal correlation. When there is no correlation, the eigenvalue distribution of the Wishart matrix is known as the Marchenko-Pastur distribution (MPD) in the double scaling limit. When there is temporal correlation, the eigenvalue distribution converges to the deformed MPD which has a longer tail and higher peak than the MPD. Here we discuss the moments of distribution and convergence to the deformed MPD for the Gaussian process with a temporal correlation. We show that the second moment increases as the temporal correlation increases. When the temporal correlation is the power decay, we observe a phenomenon such as a phase transition. When $\gamma>1/2$ which is the power index of the temporal correlation, the second moment of the distribution is finite and the largest eigenvalue is finite. On the other hand, when $\gamma\leq 1/2$, the second moment is infinite and the largest eigenvalue is infinite. Using finite scaling analysis, we estimate the critical exponent of the phase transition.