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.

Keywords: Wishart matrix, Marchenko-Pastur distribution, temporal correlation, phase transition, Equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, utilizing advanced random matrix theory (Wishart matrices, Marchenko-Pastur distribution) and complex phase transition analysis, but offers limited empirical validation with only a few financial dataset moments and no backtesting or implementation details.

  flowchart TD
    A["Research Goal<br>Determine effect of temporal correlation<br>on Wishart eigenvalue distribution"] --> B["Methodology<br>Define temporal correlation model & Wishart matrix"]
    B --> C{"Data Input<br>Time series with temporal correlation<br>(Gaussian process with power decay)"}
    C --> D["Computational Process<br>Analyze spectral moments &<br>Compute eigenvalue distribution"]
    D --> E{"Phase Transition Check<br>Temporal correlation power index γ"}
    E -- γ > 1/2 --> F["Key Finding: Finite Phase<br>Second moment finite<br>Largest eigenvalue finite"]
    E -- γ ≤ 1/2 --> G["Key Finding: Infinite Phase<br>Second moment infinite<br>Largest eigenvalue infinite"]
    F --> H["Outcome: Deformed Marchenko-Pastur Distribution<br>Longer tail & higher peak<br>Estimated critical exponent"]
    G --> H