On Spatial Processes and Asymptotic Inference under Near$Epoch ...

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Lemma B.3 Let fXi;ng be uniformly L2-NED on a random …eld f"i;ng with -mixing coe¢ cients (u; v; r) (u + v) b(r), 0. Let Sn = P i2Dn Xi;n and suppose that the NED coe¢ cients of fXi;ng satisfy P1 r=1 rd 1 (r) < 1 and kXk2+ < 1 for some > 0. Then, (a) (b) jCov (Xi;nXj;n)j kXk2+ n C1kXk2+ [h=3] d b =(2+ ) o ([h=3]) + C2 ([h=3]) where h = (i; j) and = =(2+ ). If in addition P 1 r=1 rd( +1) 1 b =(2+ ) (r) < 1, then for some constant C < 1, not depending on n jCov (Xi;nXj;n)j kXk2 V ar (Sn) C jDnj : n C3kXk2+ [h=3] d b =(4+2 ) o ([h=3]) + C4 ([h=3]) where h = (i; j) and = =(4+2 ). If, in addition, P 1 r=1 rd( +1) 1 b =(4+2 ) (r) < 1 where = =(4 + 2 ), then for some constant C < 1, not depending on n V ar (Sn) CkXk2 jDnj : Proof of Lemma B.3: (a) For any i 2 Dn and m > 0, let m i;n = E(Xi;njFi;n(m)); m i;n = Xi;n By the Jensen and Lyapunov inequalities, we have for all i 2 Dn; n; m 2 N and any 1 q 2 + : E m i;n and thus m i;n q m i;n q = EfjE(Xi;njFi;n(m))j q g EfE(jXi;nj q jFi;n(m))g = E jXi;nj q kXi;nk q kXk 2+ ; m i;n q 2 kXi;nk q 2 kXk 2+ : Thus, both m i;n and m i;n are uniformly L2+ bounded. Also, note that sup n;i2Dn m i;n 2 (m); given that the fXi;ng is uniformly L2-NED on f"i;ng and thus the NED-scaling factors can be chosen w.l.g. to be one. Furthermore, let ( m i;n) denote the -…eld generated by m i;n: Since ( m i;n) Fi;n(m), the mixing coe¢ cients of m i;n satisfy (1; 1; r) 1; r 2m (Mm d ; Mm d ; r 2m); r > 2m 22
where (u; v; r) are the mixing coe¢ cients of the input process ", since the mneighborhood of any point on D contains at most Mm d points of D for some M that does not depend on m, see the proof of Lemma A.1 of Jenish and Prucha (2009). Now, decompose Xi;n and and Xj;n as Xi;n = [h=3] i;n where h = (i; j). Then, jCov (Xi;nXj;n)j = Cov [h=3] + i;n ; and Xj;n = [h=3] [h=3] j;n + j;n ; Cov + Cov [h=3] i;n [h=3] i;n ; [h=3] j;n ; + [h=3] i;n ; [h=3] j;n [h=3] i;n [h=3] j;n + Cov + Cov + [h=3] j;n [h=3] i;n ; [h=3] i;n ; [h=3] j;n [h=3] j;n : (B.1) We will now bound separately each term on the r.h.s. of the last inequality. First, using Lemma B.2 with p = q = 2 + ; and r = (2 + )= yields the following bound on the …rst term: Cov [h=3] i;n ; [h=3] j;n 4 [h=3] i;n 4kXk 2 2+ 2+ C1kXk 2 2+ [h=3] d [h=3] j;n 2+ =(2+ ) (1; 1; [h=3]) (B.2) =(2+ ) M [h=3] d ; M [h=3] d ; h 2 [h=3] b =(2+ ) ([h=3]) where = =(2 + ). Second, the Cauchy-Schwartz inequality gives the following bound on the second and third terms: Cov [h=3] i;n ; [h=3] j;n 4 [h=3] i;n 2 [h=3] j;n Similarly, the fourth term can be bounded as: Cov [h=3] i;n ; [h=3] j;n 4 [h=3] i;n 2 [h=3] j;n Collecting (B.2)-(B.4), we have n jCov (Xi;nXj;n)j kXk2+ C1kXk2+ [h=3] d 2 2 4kXk2 ([h=3]) (B.3) 8kXk2 ([h=3]) (B.4) b =(2+ ) o ([h=3]) + C2 ([h=3]) which proves the …rst inequality. Using this inequality as well as the bounds on the sizes of the sets given in 23
Page 1 and 2: On Spatial Processes and Asymptotic
Page 3 and 4: 1 Introduction 1 Models with cross-
Page 5 and 6: tion of the NED property under tran
Page 7 and 8: tion error declines “su¢ ciently
Page 9 and 10: Proposition 3 (Doukhan, 1994) Under
Page 11 and 12: and sup n sup 1 i n We have the fol
Page 13 and 14: Dobrushin (1968) showed that weak d
Page 15 and 16: R k be a zero-mean random …eld th
Page 17 and 18: Gallant and White (1988) or Pötsch
Page 19 and 20: uniformly to Q n. A su¢ cient cond
Page 21 and 22: Furthermore, for the r > 2 as speci
Page 23: with kYi E[YijFi(s)]k 2 fi("i+j; Z
Page 27 and 28: The above CLT is in essence a varia
Page 29 and 30: Using part (a) of Lemma B.3 with Xi
Page 31 and 32: Next observe that Vns = n;s n " 1 n
Page 33 and 34: The above shows that indeed, for ea
Page 35 and 36: now follows immediately from part (
Page 37 and 38: Recalling that supn h max jDnj 1 i
Page 39 and 40: [29] Jenish, N., and I.R. Prucha (2

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