We propose a high-dimensional white noise test that captures serial correlations within and across component series without specifying an alternative model. The test statistic is a U-statistic based on sample autocovariances. Under the null, asymptotic normality is established as $p, T \to \infty$ jointly using martingale difference theory. Our approach imposes no cross-sectional independence assumption, requiring only spectral conditions on $Σ_0$. Theoretically, we link cross-sectional correlations to a graph structure, integrating algebraic and geometric analyses to facilitate the derivation. Simulations confirm reliable size control and satisfactory power across various $(p, T)$ settings.

翻译：我们提出一种高维白噪声检验方法，该方法无需设定替代模型即可捕获分量序列内及跨分量序列的序列相关性。检验统计量是基于样本自协方差的U统计量。在原假设下，我们利用鞅差理论建立了当$p, T$联合趋于无穷时检验统计量的渐近正态性。该方法不要求横截面独立性假设，仅需对$Σ_0$提出谱条件。理论上，我们将横截面关联与图结构相联系，融合代数和几何分析方法以简化推导过程。模拟实验证实，该方法在各种$(p, T)$设定下均能保持可靠的检验水平并具备令人满意的检验功效。