Asymptotic Normality of the Recursive Kernel Regression Estimate Under Dependence Conditions

For i = 1,2,…, let Xᵢ and Yᵢ be Rᵈ-valued (d ≥ 1 integer) and R-valued, respectively, random variables, and let \(Xᵢ, Yᵢ)\, i ≥ 1, be a strictly stationary and α-mixing stochastic process. Set m(x) = E(Y₁ X₁ = x), x ∈ Rᵈ, and let m̂ₙ(x) be a certain recursive kernel estimate of $m(x)$. Under suitable regularity conditions and as n → ∞, it is shown that m̂ₙ(x), properly normalized, is asymptotically normal with mean 0 and a specified variance. This result is established, first under almost sure boundedness of the Yᵢ's, and then by replacing boundedness by continuity of certain truncated moments. It is also shown that, for distinct points x₁,…,xN in Rᵈ (N ≥ 2 integer), the joint distribution of the random vector, (m̂ₙ(x₁),…,m̂ₙ(xN)), properly normalized, is asymptotically N-dimensional normal with mean vector 0 and a specified covariance function.