Bump detection in the presence of dependency: does it ease or does it load? (Q2203641)

The problem of bump detection in the presence of dependency is analyzed in this paper. Observations of a triangular array of Gaussian vectors \(Y=\mu_n+\xi_n\) are considered, with known positive definite covariance matrix \(\Sigma_n\), but an unknown mean vector \(\mu_n\) and the noise \(\xi_n\) consists of \(n\) consecutive samples of a stationary process \((Z_t)_{t\in\mathbb Z}\). The goal of the paper is to analyze how difficult is to detect abrupt changes based on observation \(Y\) when \(\mu_n\) is obtained by an equidistantly sampling. Also asymptotic lower and upper bounds are provided for the amplitude of detectable signals. The study is organized in five main sections, the first one containing the model and problem statement, the terminology necessary for results and presentation of some related works. The main results obtained under some assumptions are presented in the second section, obtaining asymptotic minimax detection boundary (Theorem 2.1) and non-asymptotic results in the case of a seemingly simpler testing problem with possible bumps that belong to a set of non-overlapping intervals. Applying Theorem 2.1 to ARMA\((p,q)\) processes asymptotic detection boundary is obtained. Also non-asymptotic results for AR\((p)\) noise are obtained in the third section. In the fourth section of the paper numerical simulations are given to examine the finite sample accuracy of the asymptotic upper bounds for the detection boundary. Based on laws of large numbers, the proofs of the main results are given in the fifth section of the paper. Also it is mentioned the supplementary material (\url{doi:10.3150/20-BEJ1226SUPP.pdf}) containing several results that are necessary for this study.