Minimizers of U-processes and their domains of attraction

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.