This article proposes a co-variance operator for Banach valued random elements using the concept of $U$-statistic. We then study the asymptotic distribution of the proposed co-variance operator along with related large sample properties. Moreover, specifically for Hilbert space valued random elements, the asymptotic distribution of the proposed estimator is derived even for dependent data under some mixing conditions.

翻译：本文利用$U$-统计量的概念，提出了一种针对Banach值随机元的协方差算子。随后，我们研究了所提出协方差算子的渐近分布及相关的大样本性质。此外，对于Hilbert空间值随机元，即使在满足某些混合条件的相依数据下，也推导出了所提出估计量的渐近分布。