Multivariate Distributions in Non-Stationary Complex Systems I: Random Matrix Model and Formulae for Data Analysis

Paper: arXiv 2412.11601

Authors: Efstratios Manolakis, Anton J. Heckens, Benjamin Köhler, Thomas Guhr

Abstract

Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant variables, especially in the presence of non-stationarity. We use a generalized scalar product between correlation matrices to clearly demonstrate this non-stationarity. Further, we present a model that we recently put forward, which captures how the non-stationary fluctuations of correlations make the tails of multivariate distributions heavier. Here, we provide the resulting formulae including Gaussian or Algebraic features. Compared to our previous results, we manage to remove in the Algebraic cases one out of the two, respectively three, fit parameters which considerably facilitates applications. We demonstrate the usefulness of these results by deriving joint distributions for linear combinations of amplitudes and validating them with financial data. Furthermore, we explicitly work out the moments of our model distributions. In a forthcoming paper we apply the model to financial markets.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous random matrix model to handle heavy-tailed multivariate distributions in non-stationary financial time series. While it includes strong theoretical derivations and initial empirical validation on NYSE data, full empirical backtesting and code implementation are deferred to a companion paper.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Model heavy-tailed multivariate distributions<br>under non-stationarity for risk assessment"] --> B["Data & Inputs:<br>Financial Market Data &<br>Empirical Correlation Matrices"]
    
    B --> C["Methodology:<br>Apply Generalized Scalar Product<br>to detect non-stationarity"]
    
    C --> D["Computational Process:<br>Develop Random Matrix Model capturing<br>fluctuations in correlations"]
    
    D --> E["Model Optimization:<br>Derive simplified formulae for Gaussian/Algebraic<br>tails (reduced fit parameters)"]
    
    E --> F["Validation:<br>Derive joint distributions for linear<br>combinations & validate against financial data"]
    
    F --> G["Key Outcomes:<br>Explicit distribution moments derived &<br>improved framework for systemic risk assessment"]