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n 1 t n n use none none drawer todisplay none with none Scan Barcodes with Mobile Phones plim (1/n). n t=1 n n X t2 = 2 none for none (0, ), (1 + i X t / n) = 1, R,. (7.39). lim E (7.40). sup E n 1 t=1 1 + 2 X t2 /n < , R. (7.41). Then 1 n X t d N (0, 2 ).. (7.42). Proof: Withou none none t loss of generality we may assume that 2 = 1 because, if not, we may replace X t by X t / . It follows from the rst part of Lemma 7.1 that exp i (1/ n).

t=1 n (1 + i X t / n). exp ( 2 /2)(1/n). X t2 exp r ( X t / n) .. (7.43). Condition (7.39) implies that n n plim exp ( 2 /2)(1/n). X t2 = exp( 2 /2).. (7.44). The Mathematical and Statistical Foundations of Econometrics Moreover, it follows from (7.38), (7.39), and the inequality r (x). x. 3 for x. < 1 that n t=1 r ( X t / n)I (. X t / n < 1) . 3 n n 3 . n t=1 X j 3 I . X t / n < 1 . n max1 t n X t X t2 p 0. (1/n) n t=1 Next, observe that n t=1 n r ( X t / n)I (. X t / n 1) . r ( X t / n) I (. X t / n 1). n t=1 I . max X t / n 1 . 1 t n r ( X t / n) .. (7.45). The result (7.45) and condition (7.38) imply that r ( X t / n)I (. X t / n 1) = 0 (7.46). P . max X t / n < 1 1. 1 t n Therefore, it follows from (7.38), (7.39), and (7.46) that plim exp n t=1 r ( X t / n) = 1.. (7.47). Thus, we can write exp i (1/ n). (1 + i X t none none / n) exp( 2 /2). (1 + i X t / n) Z n ( ),. (7.48). Dependent Laws of Large Numbers and Central Limit Theorems where Z n ( ) = ex p( 2 /2) exp ( 2 /2)(1/n). t=1 n X t2 (7.49). exp r ( X t / n). p 0.. Because Z n ( ). 2 with probability 1 given that exp( x 2 /2 + r (x)). 1, it follows from (7.49) and the dominated-convergence theorem that (7.50). lim E Z n ( ). 2 = 0. . (7.51) z z ) that Moreover, condition (7.41) implies (using zw = z w and z. = 2 n sup E (1 + i X t / n) . n 1 t=1 = sup E n 1 t=1 n (1 + i X t / n)(1 i X t / n). = sup E n 1 t=1 (1 + 2 X t2 /n) < . (7.52). Therefore, it follows from the Cauchy Schwarz inequality and (7.51) and (7.52) that n n lim E Z n ( ). (1 + i X t / n). lim E[. Z n ( ). 2 ] sup E n 1 t=1 (1 + 2 X t2 /n) = 0. (7.53). Finally, it f none for none ollows now from (7.40), (7.48), and (7.

53) that lim E exp i (1/ n). = exp( 2 /2).. (7.54). Because the r ight-hand side of (7.54) is the characteristic function of the N(0, 1) distribution, the theorem follows for the case 2 = 1 Q.E.

D.. The Mathematical and Statistical Foundations of Econometrics Lemma 7.2 is the basis for various central limit theorems for dependent processes. See, for example, Davidson s (1994) textbook.

In the next section, I will specialize Lemma 7.2 to martingale difference processes..

7.5.3.

Martin none none gale Difference Central Limit Theorems Note that Lemma 7.2 carries over if we replace the X t s by a double array X n,t , t = 1, 2, . .

. , n, n = 1, 2, 3, . .

. . In particular, let Yn,1 = X 1 ,.

Yn,t = X t I (1/n). 2 Xk 2 + 1 t 2.. (7.55). Then, by condition (7.39),. P[Yn,t = X t none for none for some t n] P[(1/n). X t2 > 2 + 1] 0; (7.56). hence, (7.42) holds if 1 n Yn,t d N (0, 2 ).. (7.57). Therefore, it suf ces to verify the conditions of Lemma 7.2 for (7.55).

First, it follows straightforwardly from (7.56) that condition (7.39) implies.

plim(1/n). n t=1 2 Yn,t = 2 . (7.58). 2 Moreover, i none none f X t is strictly stationary with an -mixing base and E[X 1 ] = 2 (0, ), then it follows from Theorem 7.7 that (7.39) holds and so does (7.

58). Next, let us have a closer look at condition (7.38).

It is not hard to verify that, for arbitrary > 0,. P max X t / n > = P (1/n). 1 t n n t=1 X t2 I (. X t / n > ) > 2 . (7.59).

Hence, (7.38) is equivalent to the condition that, for arbitrary > 0,. (1/n). X t2 I (. X t > n) p 0. (7.60). Dependent Laws of Large Numbers and Central Limit Theorems Note that (7.60) is true if X t is strictly stationary because then E (1/n). 2 X t2 I (. X t > n) = E X 1 I (. X 1 . > n) 0..
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