Testing mutual independence among multiple random variables is a fundamental problem in statistics, with wide applications in genomics, finance, and neuroscience. In this paper, we propose a new class of tests for high-dimensional mutual independence based on $L$-statistics. We establish the asymptotic distribution of the proposed test when the order parameter $k$ is fixed, and prove asymptotic normality when $k$ diverges with the dimension. Moreover, we show the asymptotic independence of the fixed-$k$ and diverging-$k$ statistics, enabling their combination through the Cauchy method. The resulting adaptive test is both theoretically justified and practically powerful across a wide range of alternatives. Simulation studies demonstrate the advantages of our method.

翻译：检验多个随机变量之间的相互独立性是统计学中的一个基本问题，在基因组学、金融学和神经科学等领域具有广泛应用。本文提出了一类基于$L$统计量的高维相互独立性检验新方法。当阶数参数$k$固定时，我们建立了所提检验的渐近分布；当$k$随维度发散时，证明了渐近正态性。此外，我们证明了固定$k$与发散$k$统计量的渐近独立性，从而可通过柯西方法将其组合。所得的自适应检验在理论上有严格依据，且对广泛的备择假设具有实际检验效力。模拟研究验证了本方法的优势。