This paper adresses the problem of testing for the equality of $k$ probability distributions on Hilbert spaces, with $k\geqslant 2$. We introduce a generalization of the maximum variance discrepancy called multiple maximum variance discrepancy (MMVD). Then, a consistent estimator of this measure is proposed as test statistic, and its asymptotic distribution under the null hypothesis is derived. A simulation study comparing the proposed test with existing ones is provided

翻译：本文研究了Hilbert空间上 $k$ 个概率分布相等性的检验问题，其中 $k\geqslant 2$。我们引入了最大方差差异的推广形式，称为多重最大方差差异（MMVD）。随后，提出了该度量的一致性估计量作为检验统计量，并推导了其在原假设下的渐近分布。通过模拟研究，将所提出的检验方法与现有方法进行了比较。