Geometric quantiles are location parameters which extend classical univariate quantiles to normed spaces (possibly infinite-dimensional) and which include the geometric median as a special case. The infinite-dimensional setting is highly relevant in the modeling and analysis of functional data, as well as for kernel methods. We begin by providing new results on the existence and uniqueness of geometric quantiles. Estimation is then performed with an approximate M-estimator and we investigate its large-sample properties in infinite dimension. When the population quantile is not uniquely defined, we leverage the theory of variational convergence to obtain asymptotic statements on subsequences in the weak topology. When there is a unique population quantile, we show, under minimal assumptions, that the estimator is consistent in the norm topology for a wide range of Banach spaces including every separable uniformly convex space. In separable Hilbert spaces, we establish weak Bahadur-Kiefer representations of the estimator, from which $\sqrt n$-asymptotic normality follows. As a consequence, we obtain the first central limit theorem valid in a generic Hilbert space and under minimal assumptions that exactly match those of the finite-dimensional case. Our consistency and asymptotic normality results significantly improve the state of the art, even for exact geometric medians in Hilbert spaces.

翻译：几何分位数是位置参数，将经典单变量分位数推广至赋范空间（可能为无限维），并以几何中位数作为特例。无限维框架在函数型数据建模分析及核方法中具有高度相关性。我们首先给出几何分位数存在性与唯一性的新结论。随后采用近似M估计量进行估计，并探究其在无限维中的大样本性质。当总体分位数定义不唯一时，我们利用变分收敛理论在弱拓扑下获取子序列的渐近陈述。当总体分位数唯一时，我们在极弱假设下证明该估计量在广泛巴拿赫空间（包括所有可分离一致凸空间）中依范数拓扑相合。在可分离希尔伯特空间中，我们建立了估计量的弱Bahadur-Kiefer表示，由此导出$\sqrt n$-渐近正态性。作为推论，我们得到了首个在一般希尔伯特空间中成立、且假设条件与有限维情形完全匹配的中心极限定理。即使在希尔伯特空间中精确几何中位数的情形，我们的相合性与渐近正态性结果也显著改进了现有最优结论。