Bump detection in the presence of dependency: Does it ease or does it load?

TL;DR

This paper establishes the fundamental limits for detecting abrupt mean changes in stationary Gaussian processes, highlighting the role of spectral density and dependency structure, supported by asymptotic and finite-sample analyses.

Contribution

It derives the asymptotic minimax detection boundary for bumps in Gaussian processes considering dependency, and provides non-asymptotic results for AR(p) processes.

Findings

Detection boundary is determined by spectral density at zero.

Non-asymptotic results validate asymptotic findings for AR(p).

Simulation confirms theoretical predictions for finite samples.

Abstract

We provide the asymptotic minimax detection boundary for a bump, i.e. an abrupt change, in the mean function of a stationary Gaussian process. This will be characterized in terms of the asymptotic behavior of the bump length and height as well as the dependency structure of the process. A major finding is that the asymptotic minimax detection boundary is generically determined by the value of its spectral density at zero. Finally, our asymptotic analysis is complemented by non-asymptotic results for AR($p$) processes and confirmed to serve as a good proxy for finite sample scenarios in a simulation study. Our proofs are based on laws of large numbers for non-independent and non-identically distributed arrays of random variables and the asymptotically sharp analysis of the precision matrix of the process.