Uniform limit theorems for wavelet density estimators

Let pn(y)=∑kα̂kϕ(y−k)+∑l=0jn−1∑kβ̂lk2l/2ψ(2ly−k) be the linear wavelet density estimator, where ϕ, ψ are a father and a mother wavelet (with compact support), α̂k, β̂lk are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p0 on ℝ, and jn∈ℤ, jn↗∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup y∈ℝ|pn(y)−Epn(y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup y∈ℝ|pn(y)−p0(y)| attains the optimal almost sure rate of convergence for estimating p0, if jn is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of pn, that is, for the stochastic processes √n(FₙW(s)-F(s))=√n∫-∞ˢ(pₙ-p₀), s∈ℝ, are proved; and more generally, uniform central limit theorems for the processes √n∫(pₙ-p₀)f, f, for other Donsker classes F of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508–539].