We study the asymptotic properties of geodesically convex $M$-estimation on non-linear spaces. Namely, we prove that under very minimal assumptions besides geodesic convexity of the cost function, one can obtain consistency and asymptotic normality, which are fundamental properties in statistical inference. Our results extend the Euclidean theory of convex $M$-estimation; They also generalize limit theorems on non-linear spaces which, essentially, were only known for barycenters, allowing to consider robust alternatives that are defined through non-smooth $M$-estimation procedures.

翻译：我们研究了非线性空间上测地凸$M$估计的渐近性质。具体而言，我们证明在成本函数的测地凸性之外仅需极少的假设条件下，即可获得统计推断中的基本性质——相合性与渐近正态性。我们的结果拓展了欧氏空间中凸$M$估计的理论；同时推广了非线性空间上的极限定理（此前本质上仅对重心已知），使得能够考虑通过非光滑$M$估计程序定义的稳健替代方法。