We analyze the stability of (strong) laws of large numbers in Hadamard spaces with respect to distributional perturbations. For the inductive means of a sequence of independent, but not necessarily identically distributed random variables, we provide a concentration inequality in quadratic mean, as well as a strong law of large numbers, generalizing a classical result of K.-T. Sturm. For the Fr\'echet mean, we generalize H. Ziezold's law of large numbers in Hadamard spaces. In this case, we neither require our data to be independent, nor identically distributed; reasonably mild conditions on the first two moments of our sample are enough. Additionally, we look at data contamination via a model inspired by Huber's $\varepsilon$-contamination model, in which we replace a random portion of the data with noise. In the most general setup, we do neither require the data, nor the noise to be i.i.d., nor do we require the noise to be independent of the data. To analyze the stability of the (non-symmetric) inductive mean with respect to data loss, data permutation, and noise, a resampling scheme is introduced, and sufficient conditions for its convergence are provided. These results suggest that means in Hadamard spaces are as robust as in Euclidean spaces. This is underlined by a small simulation study, in which we compare the robustness of means on the manifold of positive definite matrices, with means on open books.

翻译：我们分析了Hadamard空间中（强）大数定律关于分布扰动的稳定性。对于一列独立但不必同分布的随机变量的归纳均值，我们给出了二次均值意义下的集中不等式以及强大数定律，推广了K.-T. Sturm的经典结果。对于Fr\'echet均值，我们推广了H. Ziezold在Hadamard空间中的大数定律。在此情形下，我们既不要求数据独立，也不要求同分布；仅需对样本的前两阶矩施加合理的温和条件。此外，我们通过受Huber $\varepsilon$-污染模型启发的模型来研究数据污染问题，该模型将数据的随机部分替换为噪声。在最一般的设定中，我们既不要求数据与噪声独立同分布，也不要求噪声与数据独立。为分析（非对称）归纳均值对数据丢失、数据置换及噪声的稳定性，我们引入了一种重采样方案，并给出了其收敛的充分条件。这些结果表明Hadamard空间中的均值具有与欧几里得空间中相当的鲁棒性。一项小规模模拟研究进一步印证了这一点，我们在其中比较了正定矩阵流形上的均值与开书结构上的均值的鲁棒性。