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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.
62F12 Asymptotic properties of parametric estimators
60F17 Functional limit theorems; invariance principles
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