In this paper, we study the minimizers of U-processes and their domains of attraction. U-processes arise in various statistical contexts, particularly in M-estimation, where estimators are defined as minimizers of certain objective functions. Our main results establish necessary and sufficient conditions for the distributional convergence of these minimizers, identifying a broad class of normalizing sequences that go beyond the standard square-root asymptotics with normal limits. We show that the limit distribution belongs to exactly one of the four classes introduced by Smirnov. These results do not only extend Smirnov's theory but also generalize existing asymptotic theories for M-estimators, including classical results by Huber and extensions to higher-degree U-statistics. Furthermore, we analyze the domain of attraction for each class, providing alternative characterizations that determine which types of statistical estimators fall into a given asymptotic regime.

翻译：本文研究U-过程的极小值及其吸引域。U-过程广泛存在于各类统计问题中，尤其在M估计中，估计量被定义为特定目标函数的极小值。我们建立了这些极小值分布收敛的充分必要条件，确定了一类超越标准正态极限的平方根渐近性的广义正规化序列。研究表明极限分布严格属于Smirnov提出的四类分布之一。这些结果不仅扩展了Smirnov理论，还推广了现有M估计量的渐近理论，包括Huber的经典结果以及高阶U统计量的扩展。此外，我们分析了每类分布的吸引域，提供了确定统计估计量所属渐近机制的替代特征刻画。