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

Co-variance Operator of Banach Valued Random Elements: U-Statistic Approach