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

TL;DR

This paper investigates how temporal correlations in time series cause the eigenvalue distribution of Wigner matrices to deform from the classic semi-circle law, revealing phase transitions and differences in financial data.

Contribution

It introduces the concept of a deformed semi-circle law for correlated time series and analyzes phase transitions compared to the Marchenko-Pastur distribution.

Findings

Eigenvalue distribution deforms with temporal correlation

Financial time series show deviations explained by correlation effects

Phase transition observed in eigenvalue distribution behavior

Abstract

We study the eigenvalue of the Wigner random matrix, which is created from a time series with temporal correlation. We observe the deformation of the semi-circle law which is similar to the eigenvalue distribution of the Wigner-L\`{e}vy matrix. The distribution has a longer tail and a higher peak than the semi-circle law. In the absence of correlation, the eigenvalue distribution of the Wigner random matrix is known as the semi-circle law in the large $N$ limit. When there is a temporal correlation, the eigenvalue distribution converges to the deformed semi-circle law which has a longer tail and a higher peak than the semi-circle law. When we created the Wigner matrix using financial time series, we test the normal i.i.d. using the Wigner matrix. We observe the difference from the semi-circle law for FX time series. The difference from the semi-circle law is explained by the temporal correlation. Here, we discuss the moments of distribution and convergence to the deformed semi-circle law with a temporal correlation. We discuss the phase transition and compare to the Marchenko-Pastur distribution(MPD) case.