Marchenko-Pastur Distribution

Deformation of semi-circle law for the correlated time series and Phase transition ArXiv ID: 2508.07192 “View on arXiv” Authors: Masato Hisakado, Takuya Kaneko 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-Lèvy 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. ...

Correlations versus noise in the NFT market ArXiv ID: 2404.15495 “View on arXiv” Authors: Unknown Abstract The non-fungible token (NFT) market emerges as a recent trading innovation leveraging blockchain technology, mirroring the dynamics of the cryptocurrency market. The current study is based on the capitalization changes and transaction volumes across a large number of token collections on the Ethereum platform. In order to deepen the understanding of the market dynamics, the collection-collection dependencies are examined by using the multivariate formalism of detrended correlation coefficient and correlation matrix. It appears that correlation strength is lower here than that observed in previously studied markets. Consequently, the eigenvalue spectra of the correlation matrix more closely follow the Marchenko-Pastur distribution, still, some departures indicating the existence of correlations remain. The comparison of results obtained from the correlation matrix built from the Pearson coefficients and, independently, from the detrended cross-correlation coefficients suggests that the global correlations in the NFT market arise from higher frequency fluctuations. Corresponding minimal spanning trees (MSTs) for capitalization variability exhibit a scale-free character while, for the number of transactions, they are somewhat more decentralized. ...

Deformation of Marchenko-Pastur distribution for the correlated time series ArXiv ID: 2305.12632 “View on arXiv” Authors: Unknown Abstract 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 $γ>1/2$ 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 $γ\leq 1/2$, the second moment is infinite and the largest eigenvalue is infinite. Using finite scaling analysis, we estimate the critical exponent of the phase transition. ...