Minimizers of convex U-processes and their domains of attraction

D. Ferger
Theory of Stochastic Processes
Vol.29 (45), no.2, 2025, pp.1-29
In this paper, we study the minimizers of convex 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 [20]. These results do not only extend Smirnov's theory but also generalize existing asymptotic theories for M-estimators, including classical results by Huber [12] 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.
DOI: https://doi.org/10.3842/tsp-5376938787-76
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