Relevant change points in high dimensional time series

This paper investigates the problem of detecting relevant change points in the mean vector, say μₜ =(μ1,t,… ,μd,t)T of a high dimensional time series (Zₜ)t∈ Z. While the recent literature on testing for change points in this context considers hypotheses for the equality of the means μₕ⁽¹⁾ and μₕ⁽²⁾ before and after the change points in the different components, we are interested in a null hypothesis of the form H₀: |μ⁽¹⁾ₕ - μ⁽²⁾ₕ | ≤ Δₕ ~~~ for all ~~h=1,… ,d where Δ₁, … , Δd are given thresholds for which a smaller difference of the means in the h-th component is considered to be non-relevant. We propose a new test for this problem based on the maximum of squared and integrated CUSUM statistics and investigate its properties as the sample size n and the dimension d both converge to infinity. In particular, using Gaussian approximations for the maximum of a large number of dependent random variables, we show that on certain points of the boundary of the null hypothesis a standardised version of the maximum converges weakly to a Gumbel distribution.