| Summary: | We propose a two-sample testing procedure for high-dimensional time series. To obtain the asymptotic distribution of our <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>ℓ</mo><mo>∞</mo></msub></semantics></math></inline-formula>-type test statistic under the null hypothesis, we establish high-dimensional central limit theorems (HCLTs) for an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing sequence. Specifically, we derive two HCLTs for the maximum of a sum of high-dimensional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing random vectors under the assumptions of bounded finite moments and exponential tails, respectively. The proposed HCLT for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing sequence under bounded finite moments assumption is novel, and in comparison with existing results, we improve the convergence rate of the HCLT under the exponential tails assumption. To compute the critical value, we employ the blockwise bootstrap method. Importantly, our approach does not require the independence of the two samples, making it applicable for detecting change points in high-dimensional time series. Numerical results emphasize the effectiveness and advantages of our method.
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