Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers
提出了一个通用的均匀大数定律，只需添加一个平滑条件（如Lipschitz条件或导数条件），就能将逐点大数定律转化为紧集上的均匀大数定律，适用于非线性计量经济模型中参数和非参数估计量的一致性和渐近正态性证明。
A basic tool of modern econometrics is a uniform law of large numbers (LLN). It is a primary ingredient used in proving consistency and asymptotic normality of parametric and nonparametric estimators in nonlinear econometric models. Thus, in a well-known review article, Burguete, Gallant, and Sousa [8, p. 162] introduce a uniform LLN with the statement: following theorem is the result upon which the asymptotic theory of nonlinear econometrics rests. So pervasive is the use of uniform LLNs, that numerous authors appeal to an unspecified generic uniform LLN. Others appeal to some specific result. The purpose of this paper is to provide a generic uniform LLN that is sufficiently general to incorporate most applications of uniform LLNs in the nonlinear econometrics literature. In summary, the paper presents a result that can be used to turn state of the art pointwise LLNs into uniform LLNs over compact sets, with the addition of a single smoothness condition -- either a Lipschitz condition or a derivative condition. The latter is particularly easy to verify, and is implied by common assumptions used to prove asymptotic normality of estimators. Thus, the additional condition is not particularly restrictive. In contrast to other uniform LLNs that appear in the literature, the one given here allows the full range of heterogeneity of summands (i.e., non-identical distributions), and temporal dependence, that is available with pointwise LLNs.