We study a mean change point testing problem for high-dimensional data, with exponentially- or polynomially-decaying tails. In each case, depending on the $\ell_0$-norm of the mean change vector, we separately consider dense and sparse regimes. We characterise the boundary between the dense and sparse regimes under the above two tail conditions for the first time in the change point literature and propose novel testing procedures that attain optimal rates in each of the four regimes up to a poly-iterated logarithmic factor. Our results quantify the costs of heavy-tailedness on the fundamental difficulty of change point testing problems for high-dimensional data by comparing to the previous results under Gaussian assumptions. To be specific, when the error vectors follow sub-Weibull distributions, a CUSUM-type statistic is shown to achieve a minimax testing rate up to $\sqrt{\log\log(8n)}$. When the error distributions have polynomially-decaying tails, admitting bounded $\alpha$-th moments for some $\alpha \geq 4$, we introduce a median-of-means-type test statistic that achieves a near-optimal testing rate in both dense and sparse regimes. In particular, in the sparse regime, we further propose a computationally-efficient test to achieve the exact optimality. Surprisingly, our investigation in the even more challenging case of $2 \leq \alpha < 4$, unveils a new phenomenon that the minimax testing rate has no sparse regime, i.e. testing sparse changes is information-theoretically as hard as testing dense changes. This phenomenon implies a phase transition of the minimax testing rates at $\alpha = 4$.

翻译：研究重尾分布（指数衰减或多项式衰减）下高维数据的均值变点检验问题。根据均值变化向量的$\ell_0$范数，分别考虑稠密与稀疏两种情形。在变点检验文献中，首次在上述两种尾部条件下刻画了稠密与稀疏情形间的边界，并提出了四种情形下均能达到最优检验速率（至多相差多迭代对数因子）的新检验方法。通过与高斯假设下的已有结果对比，量化了重尾性对高维数据变点检验基础困难度的代价。具体而言：当误差向量服从子威布尔分布时，基于CUSUM的检验统计量可达到$\sqrt{\log\log(8n)}$量级的极小极大检验速率；当误差分布具有多项式衰减尾部（存在$\alpha \geq 4$的有界$\alpha$阶矩）时，引入的中位数均值型检验统计量在稠密与稀疏情形下均能实现近最优检验速率。特别地，在稀疏情形中进一步提出计算高效的检验方法以达到精确最优性。令人意外的是，在更具挑战性的$2 \leq \alpha < 4$情形的探究中，揭示了极小极大检验速率不存在稀疏情形的新现象，即检验稀疏变化与检验稠密变化在信息论意义上具有相同难度。该现象表明在$\alpha = 4$处极小极大检验速率发生相变。