Deformation of semi-circle law for the correlated time series and Phase transition

We study Wigner-type random matrices constructed from financial time series with temporal correlations. We characterize the deformed spectral law through its moments and observe behavior consistent with a modified semicircle law. We apply our framework to several financial time series and observe deviations from the semicircle law in specific foreign exchange markets. The difference from the semicircle law for the financial time series depends on the temporal correlation of financial time series. We provide a moment analysis for both exponential and power law correlation structures and show that the fourth moment increases with the strength of correlations. In the case of power law decay, a transition emerges between regimes with finite and divergent higher-order moments. Finally, numerical simulations support the analytical predictions and reveal finite-size scaling behavior near the transition.