Necessary and sufficient consistency conditions for a recursive kernel regression estimate

A recursive kernel estimate [summation operator]i = 1n YiK[+45 degree rule](x - Xi)hi)[+45 degree rule][summation operator]j = 1n K((x - Xj)[+45 degree rule]hj) of a regression m(x) = E{YX = x} calculated from independent observations (X1, Y1),..., (Xn, Yn) of a pair (X, Y) of random variables is examined. ForEY1 + [delta] < [infinity], [delta] > 0, the estimate is weakly pointwise consistent for almost all ([mu]) x [set membership, variant] Rd, [mu] is the probability measure of X, if and only if[summation operator]i-1n hid I{hi > [var epsilon] } [+45 degree rule] [summation operator]j = 1n hjd --> 0 as n --> [infinity], all [var epsilon] > 0, and[summation operator]i = 1[infinity] hid = [infinity], d is the dimension of X. For EY1 + [delta] < [infinity], [delta] > 0, the estimate is strongly pointwise consistent for almost all ([mu]) x [set membership, variant] Rd, if and only if the same conditions hold. ForEY1 + [delta] < [infinity], [delta] > 0, weak and strong consistency are equivalent. Similar results are given for complete convergence.