\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point.For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

翻译：我们引入球面深度这一概念，它是哈达玛流形上的一种内蕴统计深度，并将其最大化集合定义为布泽曼中位数。该构造利用了如下事实：Tukey半空间深度中的线性泛函本身是重归一化距离函数的极限；在哈达玛流形上，相同的极限过程生成布泽曼函数，其子水平集为半空间的天然替代——球体。所得深度由视觉边界参数化，具有等距等变性，且无需切空间线性化或选定基点。对于任意哈达玛流形，我们证明了深度区域具有嵌套性和测地凸性，深度至少为$1/(d+1)$的中心点存在，进而对任意Borel概率测度，布泽曼中位数存在。在严格负截面曲率及温和正则性假设下，深度严格拟凹且中位数唯一。我们还建立了鲁棒性：深度在总变差扰动下稳定；对于逃逸至无穷远的污染，极限中位数仅依赖于逃逸方向，而不依赖于污染质量沿测地线移动的距离（这与弗雷歇均值形成对比）。最后，我们建立了样本深度的一致收敛性，以及样本深度区域和样本布泽曼中位数的收敛性；在非紧型对称空间上，论证基于上半球半空间的VC分析，而在一般哈达玛流形上，则通过非原子性假设下的紧致性论证完成。