Testing of two-dimensional Gaussian processes by sample cross-covariance function.

Area of Science:

• Statistics
• Time Series Analysis
• Stochastic Processes

Background:

• Multivariate Gaussian processes are crucial in diverse applications.
• Accurate model identification for multivariate data is essential but challenging due to component dependencies.
• Existing methods for one-dimensional processes need generalization for multi-dimensional cases.

Purpose of the Study:

• To develop and validate a novel testing methodology for two-dimensional Gaussian processes with cross-dependency.
• To generalize existing one-dimensional Gaussian process testing techniques to a two-dimensional setting.
• To assess the practical applicability of the proposed method in real-world financial risk analysis.

Main Methods:

• The proposed methodology utilizes the sample cross-covariance function.
• It extends the sample autocovariance function approach for one-dimensional Gaussian processes.
• The method's efficiency is tested on simulated two-dimensional Gaussian processes (Brownian motion, fractional Brownian motion, AR(1)).

Main Results:

• The testing methodology demonstrates high effectiveness, even with limited sample sizes.
• Simulation results confirm the procedure's ability to accurately identify underlying process structures.
• The method proved successful in analyzing real-world financial time series data.

Conclusions:

• The introduced testing methodology is intuitive and simple to implement for two-dimensional Gaussian processes.
• It provides a reliable tool for model identification in multivariate time series analysis.
• The approach has broad applicability in various fields dealing with complex, dependent data structures.