The Equivalence of Weak, Strong and Complete Convergence in L₁ for Kernel Density Estimates

Let f be a density on Rᵈ, and let fₙ be the kernel estimate of f, fₙ(x) = (nhᵈ)⁻¹ ∑ⁿᵢ₌₁ K((x - Xᵢ)/h) where h = hₙ is a sequence of positive numbers, and K is an absolutely integrable function with ∫ K(x) dx = 1. Let Jₙ = ∫ |fₙ(x) - f(x)| dx. We show that when limₙh = 0 and limₙnhᵈ = ∞, then for every ε > 0 there exist constants r, n₀ > 0 such that P(Jₙ ≥ ε) ≤ exp(-rn), n ≥ n₀. Also, when Jₙ → 0 in probability as n → ∞ and K is a density, then limₙh = 0 and limₙnhᵈ = ∞.