Distribution-Free Pointwise Consistency of Kernel Regression Estimate

An estimate ∑ⁿᵢ₌₁ YᵢK((x - Xᵢ)/h)/∑ⁿⱼ₌₁ K((x - Xⱼ)/h), calculated from a sequence (X₁, Y₁), ⋯, (Xₙ, Yₙ) of independent pairs of random variables distributed as a pair $(X, Y)$, converges to the regression E X = x\ as n tends to infinity in probability for almost all (μ) x ∈ Rᵈ, provided that E|Y| < ∞, h → 0 and nhᵈ → ∞ as n → ∞. The result is true for all distributions μ of X. If, moreover, |Y| ≤ γ < ∞ and nhᵈ/log n → ∞ as n → ∞, a complete convergence holds. The class of applicable kernels includes those having unbounded support.