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1. : (B.2) Proof. Let ^ f (i) Y (y) for i = 1; 2 be the ith derivative of the kernel marginal density estimator. Using standard methods for kernel estimators (c.f. Robinson, 1983), we obtain under the assumptions of the lemma that, as n ! 1; h ! 0, and nh1+2i ! 1, p nh1+2i ^ f (i) Y (y) f (i) Y (y) h2 2f (i+2) Y (y) !d N (0; Vi (y)) (B.3) where Vi (y) = fY (y) R
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23. fX (U (y) ; 0) p nh ^ fY (y) fY (y) h2 2f (2) Y (y) + oP (1) : Using (B.3) and the same arguments as in Kristensen (2011, Proof of Theorem 1), we arrive at (B.1). Meanwhile, from (4.3) we have ^ U00 (y) = ^ f0 Y (y) fX( ^ U (y) ; ^) f0 X( ^ U (y) ; ^) ^ fY (y)2 fX( ^ U (y) ; ^)3 : De…ne ~ U00 (y) = ^ f0 Y (y) fX (U (y) ; 0) f0 X (U (y) ; 0) fY (y)2 fX (U (y) ; 0)3 ; and a similar argument leads to p nh3 ^ U00 (y) U00 (y) h2 2f (3) Y (y) fX (U (y) ; 0) = p nh3 ~ U00 (y) U00 (y) h2 2f (3) Y (y) fX (U (y) ; 0) + op (1) = fX (U (y) ; 0) p nh3 f0 Y (y) f0 Y (y) h2 2f (3) Y (y) + op (1) which together with (B.3) yield (B.2).
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43. Proof of Theorem 4.1. We …rst note that the PMLE takes the same form as the one analyzed in Chen and Fan (2006) with the general copula considered in their work satisfying eq. (2.13). The desired result will follow if we can verify that the conditions stated in their proof are satis…ed by our assumptions: First, by Assumptions 2.1, the discrete sample fXi : i = 0; 1; : : : ; ng generated by the UPD X is …rst-order Markovian and has absolutely continuous marginal distribution FX (x; ), marginal density fX (x; ) and transition density pX (xjx0; ) with respect to the Lebesgue measure. Hence, the copula density cX (u0; u; ) in (2.13) implied by X is absolutely continuous with respect to the Lebesgue measure on [0; 1]2 due to its continuity in FX (x; ), fX (x; ) and pX (xjx0; ).
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44. R K(i) (z)2 dz. Assumptions 2.1 and 4.4 ensure that fY (y) is su ciently smooth so that f (2) Y (y) and f (3) Y (y) exist. Assumption 4.2(i) and 4.6 regulate the mixing property of Y and the kernel function, respectively, as required by Robinson (1983). From (4.2) we have ^ U0 (y) = ^ fY (y) =fX( ^ U (y) ; ^). Now de…ne ~ U0 (y) = ^ fY (y) =fX(U (y) ; 0) and note that Assumption 4.4 and 4.5 together with the delta-method implies ^ U0 (y) ~ U0 (y) = OP (1= p n) =oP (1= p nh). It then follows that p nh ^ U0 (y) U0 (y) h2 2f (2) Y (y) fX (U (y) ; 0) = p nh ^ U0 (y) ~ U0 (y) + ~ U0 (y) U0 (y) h2 2f (2) Y (y) fX (U (y) ; 0) = p nh oP 1= p nh + ~ U0 (y) U0 (y) h2 2f (2) Y (y) fX (U (y) ; 0) = p nh ~ U0 (y) U0 (y) h2 2f (2) Y (y) fX (U (y) ; 0) + oP (1) =
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