Consistency of hh-mode depth

•Consistency of the hh-mode depth for general Banach-valued data is established.•Rates of convergence of the hh-mode depth process are found.•Qualitative robustness property of the depth is proved.•Consistency results for the associated modes of general distributions are provided.

In this paper we establish consistency results for the sample hh-mode depth in the general case of Banach-valued data. The rate of convergence is provided, which is linked to the rate at which the sample sequence of bandwidths converges to its population version. The robustness of the hh-mode depth, as well as the convergence of the associated modes of the distribution, is also studied.