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- Proof of Lemma 3: This result can be proved using similar arguments as in the proof of Lemma 1 in Kong et al. (2010). Lemma 4. Suppose that Assumptions 2.1-2.4 hold. Then there is a constant C > 0 such that, for each M > 0 and all large n, sup z∈Z sup |α|≤M d (1) n |β|≤M1/4d (2) n n X i=1 E Ωn,i,j,Ä (z; α, β) − nh 2 α(α + 2β)Sn,j(z) ≤ CM3/2 dn1, where dn1 = (nh)−3/4 (ln n)7/4 .
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- Wj(Xi, Di)I{qj,Ä (z) − Acn ≤ Yi ≤ qj,Ä (z) + Acn} : z ∈ Z is also Euclidean, which together with ln n nhcn = o(1) implies the conditions required by Theorem II.37 of Pollard (1984) are met. In addition, by straightforward calculations, we know that E K (Zi − z)/h Wj(Xi, Di)I{qj,Ä (z) − Acn ≤ Yi ≤ qj,Ä (z) + Acn} 2 = E K2 (Zi − z)/h p(Xi)−j (1 − p(Xi))j−1 E h
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