# Relevant change points in high dimensional time series

## TL;DR

This paper develops a new statistical test for detecting relevant change points in the mean vectors of high-dimensional time series, accounting for thresholds to distinguish relevant from non-relevant changes, and analyzes its asymptotic properties.

## Contribution

It introduces a novel test based on maximum CUSUM statistics for relevant change points in high-dimensional data, with theoretical analysis of its asymptotic distribution.

## Key findings

- The test converges to a Gumbel distribution under certain conditions.
- It effectively distinguishes relevant from non-relevant mean changes.
- The method is applicable as both sample size and dimension grow large.

## 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.