Title: Relevant change points in high dimensional time series,
Abstract: This paper investigates the problem of detecting relevant change points in
the mean vector, say $\mu_t =(\mu_{1,t},\ldots ,\mu_{d,t})^T$ of a high
dimensional time series $(Z_t)_{t\in \mathbb{Z}}$.
While the recent literature on testing for change points in this context
considers hypotheses for the equality of the means $\mu_h^{(1)}$ and
$\mu_h^{(2)}$ before and after the change points in the different components,
we are interested in a null hypothesis of the form $$ H_0: |\mu^{(1)}_{h} -
\mu^{(2)}_{h} | \leq \Delta_h ~~~\mbox{ for all } ~~h=1,\ldots ,d $$ where
$\Delta_1, \ldots , \Delta_d$ 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.