A Second Order Cumulant Spectrum Test That a Stochastic Process is Strictly Stationary and a Step Toward a Test for Graph Signal Strict Stationarity

TL;DR

This paper introduces a frequency domain statistical test based on second order cumulant spectrum to determine if a stochastic process is strictly stationary, with potential applications in graph signal analysis and neural network learning.

Contribution

It develops a novel test for strict stationarity using the second order cumulant spectrum in the frequency domain, including derivation, properties, and real data application.

Findings

Test statistic follows an asymptotic complex normal distribution.

Successfully applied to gamma ray decay data.

Lays groundwork for testing stationarity of graph signals.

Abstract

This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process in the frequency domain. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs in the principal domain. The test uses a window averaged sample estimate of the second order cumulant spectrum to build a test statistic with an asymptotic complex standard normal distribution. We derive the test statistic, study the properties of the test and demonstrate its application using 137Cs gamma ray decay data. Future areas of research include testing for strict stationarity of graph signals, with applications in learning convolutional neural networks on graphs, denoising, and inpainting.