Uniform laws of large numbers and stochastic Lipschitz-continuity
Let \((k_n)_{n\geq 1}\) be a sequence of positive integers and \(\Theta_n\) a subset of \(\mathbb{R}^{k_n}\) for any \(n\geq 1\). The author establishes uniform laws of large numbers for classes of real-valued functions \(\{f_n(X_k, \theta):\theta \in\Theta_n\), \(1\leq k\leq n\}\), \(n\geq 1\), obeying a stochastic Lipschitz continuity hypothesis. Here \((X_k)_{k\geq 1}\) is a sequence of independent real-valued random variables. Conditions are given under which \[ \sup_{\theta \in\Theta_n} \left|n^{-1}\sum^n_{k=1} \bigl(f_n(X_k, \theta) -Ef_n (X_k,\theta) \bigr)\right |\to 0 \] in probability or almost surely as \(n\to\infty\). The results obtained are used to derive two weak consistency results for nonlinear least-squares estimators. For related work see \textit{D. W. K. Andrews} [Econometrica 55, 1465-1471 (1987; Zbl 0646.62101); Econ. Theory 8, 241-257 (1992)], \textit{W. K. Newey} [Econometrica 59, No. 4, 1161-1167 (1991; Zbl 0743.60012)] and \textit{B. M. Pötscher} and \textit{I. R. Prucha} [ibid. 57, No. 3, 675-683 (1989; Zbl 0685.62091); Econ. Rev. 10, No. 2, 125-216 (1991; Zbl 0737.62096)]; J. Econom. 60, No. 1/2, 23-63 (1994; Zbl 0788.60042)].
scientific article; zbMATH DE number 1342869
scientific article