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Asymptotic suprema of averaged random functions. (English) Zbl 0745.62022
Summary: Suppose $$X_ i$$ are i.i.d. random variables taking values in $${\mathcal X}$$, $$\Theta$$ is a parameter space and $$y: {\mathcal X} \times \Theta \to {\mathbf R}$$ is a map. Define the averages $S_ n(y,\theta) = \left( {1 \over n} \right) \sum_{i=1}^ n y(X_ i,\theta)$ and the truncated expectations $T_ m(y,\theta) = {\mathbf E}\max \bigl(y(X_ 1,\theta),- m\bigr).$ Under the hypothesis of global dominance [i.e., $${\mathbf E}\sup_ \Theta y(X_ 1,\theta) < \infty$$] and some regularity conditions, the main result of the paper characterizes the asymptotic suprema of $$S_ n$$ as follows. For any subset $$G$$ of $$\Theta$$, with probability 1, $\lim_{n\to\infty} \sup_{\theta\in G} S_ n(y,\theta) = \lim_{m\to \infty} \sup_{\theta\in G} T_ m(y,\theta).$ This has immediate application to consistency of M- estimators. In particular, under global dominance, maxima of $$S_ n$$ must converge to the same limit as the maxima of $$T_ m(y,\theta)$$ almost surely. We also obtain necessary and sufficient conditions for consistency resembling Huber’s in the case of local dominance [i.e., each $$\theta \in \Theta$$ has a neighborhood $$N(\theta)$$ such that $${\mathbf E}\sup_{\psi \in N(\theta)} y(X_ 1,\psi) < \infty$$]. In this case there must exist a function $$b(\theta) \geq 1$$ such that $$y/b$$ is globally dominated and maxima of $$T_ m(y/b,\theta)$$ converge.
##### MSC:
 62F12 Asymptotic properties of parametric estimators 60F17 Functional limit theorems; invariance principles
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