ASYMPTOTIC NORMALITY OF THE RECURSIVE KERNEL REGRESSION ESTIMATE UNDER DEPENDENCE CONDITIONS

For i = 1, 2,..., let X(i) and Y(i) be R(d)-valued (d greater-than-or-equal-to 1 integer) and R-valued, respectively, random variables, and let {(X(i), Y(i))}, i greater-than-or-equal-to 1, be a strictly stationary and alpha-mixing stochastic process. Set m(x) = E(Y1\X1 = x), x is-an-element-of R(d), and let m(n)(x) be a certain recursive kernel estimate of m(x). Under suitable regularity conditions and as n --> infinity, it is shown that m(n)(x), properly normalized, is asymptotically normal with mean 0 and a specified variance. This result is established, first under almost sure boundedness of the Y(i)'s, and then by replacing boundedness by continuity of certain truncated moments. It is also shown that, for distinct points x1,..., x(N) in R(d) (N greater-than-or-equal-to 2 integer), the joint distribution of the random vector, (m(n)(x1),..., m(n)(x(N))), properly normalized, is asymptotically N-dimensional normal with mean vector 0 and a specified covariance function.