Supplement to ”Consistency and Asymptotic Normality of Sieve ML Estimators Under Low-Level Conditions”

This online supplement to Bierens (2013) contains the omitted proofs. Throughout I will use the same notations as in Bierens (2013), as follows. The indicator function is denoted by I(.), and N and N0 denote the sets of positive and nonnegative integers, respectively. The partial derivative to a parameter with index k will be denoted by ∇k, and ∇k,m denotes the second partial derivatives to parameters with indices k and m. To distinguish infinite dimensional parameters from finite dimensional ones, the former are displayed in bold face. Following Billingsley (1968), I will use the double-arrow ”⇒” to indicate weak convergence of sequences of random function in the metric space C[0, 1] of continuous real functions on [0, 1], endowed with the metric sup0≤u≤1 |f(u)− g(u)|, and following van der Vaart (1998), the wiggling arrow ” ” indicates weak convergence of a sequence of random elements in a Hilbert space. Finally, the operator πn applied to an infinite sequence δ = {δm}m=1 replaces all the δm’s for m > n by zeros.