Prove Lemma 24.2. Hint: First show that                          n i=1 mi(x)Ki(x, Xi) n i=1 Ki(x, Xi) − m(x) 2 µ(dx) → 0 a.s. by the use of (24.3), (24.10), (24.4), (24.8), the Toeplitz lemma, and...

Prove Lemma 24.2. Hint: First show that

by the use of (24.3), (24.10), (24.4), (24.8), the Toeplitz lemma, and Lebesgue’s dominated convergence theorem. Then formulate a recursion for {Un} with

                                 Un(x) = n i=1(Yi − mi(x))Ki(x, Xi) n i=1 Ki(x, Xi)

and show a.s. convergence of ‑ Un(x) 2µ(dx) by the use of (24.12), (24.13), and Theorem A.6 distinguishing cases (24.14) and (24.15). These results yield a.s. convergence of ‑ (mn(x) − m(x))2µ(dx) and thus, by Lemma 24.1, the assertion of part (a). Prove part (b) analogously by taking expectations.