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

More documents

Recommendations

Info

We next show that for su¢ ciently large s; By (B.9), 0 < lim inf n!1 jDnj 1 2 n;s: B 1=2 inf jDnj 1=2 n Since lims!1 (s) = 0; there exists s such that in light of (B.10) for all s s , Hence by (B.8) for all s s ; jDnj 1=2 en;s C 1=2 (s) B 1=2 =2: (B.14) jDnj 1=2 ( n en;s) jDnj 1=2 n;s and thus infn jDnj 1=2 n;s infn jDnj 1=2 n sup n jDnj 1=2 en;s. Using (??) and (B.14), we have Thus, for all s s , lim inf n!1 jDnj 1=2 1=2 B1=2 B1=2 n;s B = > 0 2 2 1 n;s X i2Dn s i;n =) N(0; 1) as n ! 1: (B.15) Since the …rst s terms do not a¤ect the analysis below we take in the following s = 1: P 1 n i2Dn Yi;n 4. CLT for Finally, using Lemma B.1 we now show that, given the maintained NED assumption, the just established CLT in (B.15) for the approximators be carried over to the the Yi;n. De…ne s i;n can X X X Wn = 1 n i2Dn Yi;n; Vns = 1 n i2Dn s i;n, Wn Vns = 1 n so that we can exploit Lemma B.1 to prove that X Wn = Yi;n =) V N(0; 1): 1 n i2Dn i2Dn We …rst verify condition (iii) of Lemma B.1. By Markov’s inequality and (B.11), for every > 0 we have lim lim sup P (jWn Vnsj > ) = lim lim sup P ( s!1 n!1 s!1 n!1 lim lim sup s!1 n!1 28 e 2 n;s = 0: 2 2 n 1 n X i2Dn s i;n s i;n > )
Next observe that Vns = n;s n " 1 n;s X i2Dn We proceed to show Wn =) V by contradiction. For that purpose let M be the set of all probability measures on (R;B), and observe that we can metrize M by, e.g., the Prokhorov distance d(:; :). Let n and be the probability measures corresponding to Wn and V , respectively, then Wn =) V , or n =) , i¤ d( n; ) ! 0 as n ! 1. Now suppose n does not converge to . Then for some > 0 there exists a subsequence fn(m)g such that d( n(m); ) > for all n(m). By (B.13), we have 0 n;s= n C < 1 for all s; n 1. Hence, 0 n(m);s= n(m) C < 1 for all n(m). Consequently, for s = 1 there exists a subsubsequence fn(m(l1))g such that n(m(l1));1= n(m(l1)) ! p(1) as l1 ! 1. For s = 2, there exists a subsubsubsequence fn(m(l1(l2)))g such that n(m(l1(l2)));2= n(m(l1(l2))) ! p(2) as l2 ! 1. The argument can be repeated for s = 3; 4:::. Now construct a subsequence fnlg such that n1 corresponds to the …rst element of fn(m(l1))g, n2 corresponds to the second element of fn(m(l1(l2)))g, and so on, then lim l!1 nl;s nl s i;n for s = 1; 2; : : : Given (B.15), it follows that as l ! 1 Then, it follows from (B.12) that lim jp(s) 1j lim s!1 s!1 lim l!1 p(s) # Vnls =) Vs N(0; p 2 (s)): : = p(s) (B.16) nl;s nl + lim s!1 sup n 1 n;s n 1 = 0: Thus Vs =) V and thus by Lemma B.1 Wnl =) V N(0; 1) as l ! 1. Since fnlg fn(m)g this contradicts the assumption that d( n(m); ) > for all n(m). This completes the proof of the CLT. Q.E.D. Proof of Corollary 1: To prove the theorem, we apply the Cramer-Wold device, and verify that for every 2 Rk P 1 with j j = 1, n Vi;n ) N(0; 1), where Vi;n = 0 Yi;n. Observe that using the properties of norms, we have and jVi;nj = 0 Yi;n j j jYi;nj = jYi;nj 1(jVi;nj > k) = 1( 0 Yi;n > k) 1(jYi;nj > k); and thus limk!1 sup n;i2Dn E[jVi;nj 2+ 1(jVi;nj > k)] = 0. Furthermore, observe that kVi;n E(Vi;n j Fi;n(s))k 2 j j kYi;n EYi;n j Fi;n(s))k 2 di;n (s) 29
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 and 24: with kYi E[YijFi(s)]k 2 fi("i+j; Z
Page 25 and 26: where (u; v; r) are the mixing coe
Page 27 and 28: The above CLT is in essence a varia
Page 29: Using part (a) of Lemma B.3 with Xi
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

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?