NECESSARY AND SUFFICIENT CONDITION FOR ASYMPTOTIC NORMALITY OF STANDARDIZED SAMPLE MEANS

NECESSARY AND SUFFICIENT CONDITION FOR ASYMPTOTIC NORMALITY OF STANDARDIZED SAMPLE MEANS BY RAJESHWARI MAJUMDAR* AND SUMAN MAJUMDAR University of Connecticut and University of Connecticut The double sequence of standardized sample means constructed from an infinite sequence of square integrable independent random vectors in the plane with identically distributed coordinates is jointly asymptotically Normal if and only if the Cesaro means of the sequence of cross sample correlation coefficients converges to!. 1. Introduction and Results. We investigate the joint asymptotic Normality of the standardized sample means, based on two random samples from two distributions, in this paper. Hereinafter, iid will abbreviate independent and identically distributed. Let \ À 4 iid sequence of random variables, mean. and variance 1 ß4 \ À 4 iid sequence of random variables, mean. and variance. 2 2ß4 In what follows, we always assume that the index 3 runs from to. Let 8 \ \ ] 8 3 3 8 3 3. 3 34. 3 œ œ 3 8 3 3 4œ 3 3 denote the standardized sample mean from the 3 th sample. Let F denote the standard Normal distribution on the line and F F the product measure on the plane, the bivariate standard Normal distribution. Our objective is to obtain necessary and sufficient conditions under which, a s 8 ß8 Ä, the double sequence of random vectors ] œ ] ß] converges in distribution to F F. 4 8ß8 8 8 Let < 3 denote the characteristic function (CF, hereinafter) of the standardized \ 3ß4, 3 3 which, by 1 and 2, does not depend on 4, and 0 8 that of ] 8 ; note that 3? 08 3? œ < 3 3 8. Since, for every? d, BÈexp3?B is bounded and continuous, where u œ, by the Central Limit Theorem (CLT, hereinafter) [Dudley (1989, Theorem 9..6)] and the definition of convergence in distribution, with ' denoting the CF of F so that '? œ exp? Î, MSC 2010 subject classifications: 60E10, 60F0. Keywords and phrases: Asymptotic normality, Levy Continuity Theorem, Lindeberg Central Limit Theorem, Net and subnet, One point compactification. 3 8 3 3

MAJUMDAR AND MAJUMDAR 2 Let < 8ß8 denote the CF of ] 8ß8 Assume that 0? œ '?. 6 8Ä 83 3 3 and < that of F F, so that < =ß > œ exp = > Î œ ' = ' >. the sequence \ À4 is independent of the sequence \ À4. 7 8ß8 8 8 ß4 ß4 By 7, < =ß> œ 0 = 0 >. Since < =ß> œ ' = ' >, < 8ß8=ß > < =ß > ' Ÿ 0 = 0 > 0 = > 0 = ' > ' = ' > ; 8 8 8 8 since every CF is bounded, by 6, for every =ß > d, < < 8ß8 Ä 8ß8 =ß > œ =ß >. 8 If 8 œ 8, then 4 follows from 8 by the Levy Continuity Theorem (LCT, hereinafter) [Dudley (1989, Theorem 9.8.2)]. Does 8 imply 4 even without the restriction 8 œ 8? The answer is yes, but to substantiate that assertion we have to come up with an appropriate LCT. We do that by: restating 4 in terms of weak convergence of induced measures, recognizing that a double sequence is a net, and formulating the LCT for nets of random vectors. The edifice thus constructed is used in establishing the main result, Theorem 1, which weakens the collection of assumptions 1, 2, and 7 that is sufficient for 4 to a collection of assumptions that includes a constituent which, in the presence of the remaining constituents, is necessary and sufficient for 4. For a separable metric space f, let Uf denote the Borel -algebra of f and `f the set of probability measures on Uf. Endowed with the topology of weak convergence, `f is metrizable as a separable metric space [Parthasarathy (1967, Theorem II.6.2)]. Since all the random elements under consideration are Borel measurable, convergence in distribution is equivalent to weak convergence of the induced probability measures [van der Vaart and Wellner (1996, page 18)]. Let Á denote the set of natural numbers; then Á Á is a directed set under the partial ordering defined by 8ß8 7ß7 Í8 7 and 8 7. A double sequence B8ß8 À 8 ß 8 taking values in f converges to B W as 8ß8 Ä if and only if the corresponding net B À Á Á converges to B. Remark 1 formulates the LCT for nets of random vectors.

3 ASYMPTOTIC NORMALITY Remark 1 Example 1.3. of van der Vaart and Wellner (1996) assert, without a proof, the LCT for any d -valued net of random vectors. They remark that a proof can be based on Prohorov's theorem [van der Vaart and Wellner (1996, Theorem 1.3.9)] and the uniqueness of CFs [Dudley (1989, Theorem 9..1)]. The proof of the LCT in the case of sequences first establishes that pointwise convergence of CFs implies uniform tightness of the underlying measures (using the Dominated Convergence Theorem) and then uses Prohorov's theorem and the uniqueness of CFs. However, it is not immediately clear how the Dominated Convergence Theorem and consequently, the proof of pointwise convergence of CFs implying uniform tightness in the case of sequences, generalizes to the case of netss. As such, we will use the LCT for nets only after verifying uniform tightness of the underlying measures. // Remark 2 completes the process of establishing that the collection of assumptions 1, 2, and 7 is sufficient for 4, by verifying the appropriate uniform tightness that would allow us to conclude 4 from 8. For Á, let \ denote \. Let 83 À Á Á È Á denote the order preserving and cofinal map that maps to its 3 th coordinate 8 3. 3 3 Remark 2 For Á Á, let ] denote ] 8 3 defined in 3 and L `d the measure induced by ] œ ] ß]. Then 4 is equivalent to (in `d ) 3ß 4œ L œ F F. 9 Recall that < œ < 8 ß8 is the CF of ], so that the assertion of 8 can be restated as < < =ß > œ =ß > 10 for every =ß > d. Once we show L À Á Á is uniformly tight, 11 by Remark 1, the LCT for nets can be used to conclude that 10 implies 9. Note that, since the kernel of the CF is bounded and continuous, 9 implies 10, so that post successful verification of 11, 9 and 10 are equivalent. 3 For Á and 3œß, let T `d denote the measure induced by \ 3ß. 3 Î 3, which, under 1 or 2, converges to F in ` d by the CLT. By 3 Proposition 9.3.4 of Dudley (1989), T À Á is uniformly tight. Consequently, 3 3 T8 À T À 3 Á Á, being contained in Á, is uniformly tight as well. By Tychonoff's theorem and Bonneferroni's inequality, 11 follows. Note that 11 is obtained without 7, that is, by using only 1 and 2. // 3ß4

MAJUMDAR AND MAJUMDAR 4 We now start the process of stating Theorem 1. The triplet of assumptions 1, 2, and 7 is equivalent to the pair of assumptions \ ß4 ß\ ß4 À 4 iid sequence of random vectors 12 We first weaken the assumption in 12 to for every 4, \ß4 and \ß4 are independent. 13 \ ß4 ß\ ß4 À 4 is an independent sequence of random vectors. 14 Let us introduce some notations here. Let 0 À Á È Á Á denote the order preserving and cofinal map given by 0 œ ß. For Á Á, let 8 œ min8 ß8 and 7 œ 8 8. Let 3 4 denote the correlation coefficient between \ ß4 and \ ß, 1 and for Á Á, let 8 3 œ 3. 16 7 44 4œ We now weaken the assumption of independence in 13 to either of Next, we introduce the assumption 3 œ! 17 3 0 œ!. 18 Js œ F, 19 where Js `d is the measure induced on Udby \ß8 \. ß8. [ s œ œ ] ß@, 20 @ œ @ ß @ d, and 8 8 3 3 @ œ!ß. 21 / 3 / 3 Theorem 1 If 1, 2, and 14 hold, then 17, 18, 9, and 19 are equivalent.

ASYMPTOTIC NORMALITY Remark 3 The collection of assumptions 1, 2, 14, and 18 is weaker than the pair of assumptions 12 and 13. To see that, consider a pair of dependent but uncorrelated random variables and a sequence of iid copies of the resulting random vector. // We present the proof of Theorem 1, an important corollary to it [Proposition 1], and the auxiliary results used in the proof of Theorem 1 in Section 2. The technical results used in all the proofs of this section are assembled in two Appendices, A.1 and A.2. 2. Proofs. We prove Theorem 1 by showing By Lemma A.1, 17 Ê 18. 17 Ê 18 Ê 9 Ê 19 Ê 17. 22 The proofs of the remaining assertions in 22 make critical use of the compactness of!ß, the one-point compactification of Ò!ß Ñ [Dudley (1989, Theorem 2.8.1)]. Let / œ 8 Î8. Since every net taking values in a compact set has a convergent subnet [Lemma A.3 ], every subnet / 9 À Y of / À Á Á has a further subnet / 9: À such that / œ,!ß. 23 9: For subsequent use, let F F3 `d denote the Normal distribution with means!, variances, and correlation coefficient 3 ß, so that F F! œ F F, and a) ` d the Normal distribution with mean! and variance ) 0, so that a! is the point mass at! and a œ F. Proof of 18 Ê 9: Recall from Remark 2 that, under 1 and 2, 9 and 10 are equivalent. Lemma 1 is a key step in proving 18 implies 10. Lemma 1 Let 3 be as in 16. If 14 holds, then, for every subsequence L0 < À < Á of L0 À Á, there exists a further subsequence L0 <7 À 7 Á such that 0 3 3 7 <7 œ ß 24 and L œ Þ 7 <7 F F 2 0 3 The proof of Lemma 1 is presented at the end of the section.

MAJUMDAR AND MAJUMDAR 6 By Lemma 1, 18 implies L0 œ F F, 26 reducing the proof to showing 26 Ê 10. Clearly, 26 implies, for every?ß@ d, < 0?ß @ œ <?ß @. 27 Fix =ß > d arbitrarily. By Lemma A.2, 10 follows if given any subnet < 9 =ß > À Y of < =ß > À Á Á, we can find a further subnet < =ß > À such that < =ß > converges to < =ß >. For 9: 9: so that Á Á, define ß if 8 8 ß if 8 8 N œ and N œ! otherwise! otherwise, ß ß ß N N œ if 8 œ 8! otherwise; ß note that N À Á Á and N À Á Á are two nets in!ß. A straightforward algebraic calculation, using 14 and, leads to the decomposition where < =ß > œ < =ß > < =ß > < =ß >, 28 ß ß < =ß > œ < =ß > N N 08 < < 08 = =ß > œ / =ß > < N 8 ß 08 < =ß > œ < 8 8 8 8 8 > =ß / > < N ß. 29 By Lemma A.3, given an arbitrary subnet < 9 À Y of < À Á Á, we ß can find a further subnet / À of / À Y, N À of ß ß 9: 9: 9 9: N À Y, and N À of N À Y such that 9 9 and 23 holds. ß ß ß ß N 9: œ N!ß and N 9: œ N!ß exist, 30 ß

7 ASYMPTOTIC NORMALITY Since 8 is order preserving and cofinal, by 29, 27, and Lemma A.1. ß ß 9: < =ß > œ < =ß > N N 31 ß By considering the two cases N œ! and N œ separately, we now show that ß ß ß 9: < =ß > œ < =ß > N. 32 If N œ!, CFs being bounded, 32 holds by 29 and 30. If N œ, then 89: 8 9: œ œ,. 33 8 9: 8 9: Since 8 is order preserving and cofinal, using, 6, and Lemma A.1, ß 8 9: < = 9: œ 8 ' =, implying by 33 and the continuity of exponential and logarithm, < 8 9: 8 =, = œ exp. 34 8 9:, 9: By 11 and Lemma A., < 0 À Á is uniformly equicontinuous; since 8 is order preserving and cofinal, by 27 and Lemma A.1, exp < 08 9 : 9: <, = > / =ß> œ =ß> œ,. 3 ß If N œ, 32 follows by 29, 34, and 3. By the same argument that established 32, with the modification that N œ implies we obtain 89: 8 9: œ 8 9: 8 9: ß 9: œ,, < =ß > œ < =ß > N. 36 Now 8 follows from 28, 31, 32, and 36. Proof of 9 Ê 19 : By Lemma A.2, it suffices to show that given an arbitrary subnet Js À Js À Js 9 Y of Á Á, there exists a further subnet 9: À such that ß

MAJUMDAR AND MAJUMDAR 8 Js 9: œ F. 37 We will establish 37 for the subnet indexed by the directed set obtained in 23. From the definition of and 3 @ in 21, we obtain @ @ œ 38 3 3 + 3 œ @ œ!ß 9:, 39, 3,3 with, œ! implying + œ and + œ!, and, œ implying + œ! and + œ. By 20, the Cauchy-Schwartz inequality, 38, and 11, Js À Á Á is uniformly tight. By the LCT for nets, 37 will follow once we show, for every? d, ' 9: where is the CF of Js Þ By 11 and Lemma A., ' since, by 20, we obtain 9:? œ '?, 40 < À Á Áis uniformly equicontinuous; 41 ' <? œ?@ ß?@, 42 '? œ <?+ ß?+ from 9 (equivalently, 10 ), 39, and 41, whence 40 follows from the definition of <, 39 and 38. Proof of 19 Ê 17 : Lemma 2 Assume 19 and Lemma 2 is a key step in this proof. Ks œ F, 43 where Ks `d denotes the measure on Udinduced by \ß8 \. ß8. Ys œ. 8 8 Then 17 holds if, with 3 4 as in 1, 3 4 œ! for 4Á. 44 The proof of Lemma 2 is presented at the end of the section.

9 ASYMPTOTIC NORMALITY Since 14 Ê 44, by Lemma 2 it suffices to show 19 Ê 43. Let @ œ @ ß@ ; s then, as in 20, Y œ ] ß@, and the argument used to establish 9 Ê 19 can be repeated verbatim to conclude 9 Ê 43. Since we have already proved 18 Ê 9, all that remains to show is 19 Ê 18. Let 3 be an arbitrary subsequence of 0< À< Á 30 À Á. It suffices to find a further subsequence 3 0<7 À7 Á that converges to!. By Lemma 1, there exists a further subsequence L0 <7 À 7 Á such that 24 and 2 hold. We are going to show that 3 of 24 equals!. Since / œ, we obtain by 21 that 7 0<7 Á Clearly, 2 implies 3 7 @ œ 0<7 3 Á. 4 =ß > œ =ß > 7 Á < 0<7 < 3, 46 where < is the CF of F F, that is, 3 3 By 42, 41, 4, and 46, < 3 =ß > œ exp = => 3 > Î. 47? œ exp? Î 7 Á ' 0<7 3, 48 where, with 3 as in 24, 3 œ 3. 49 Since RHS48 is the CF of a 3 at?, by the LCT again, 7 J 0<7 Á œ a 3 ; 0 since 7È0 <7 is order preserving and cofinal, by 19 and Lemma A.1, a 3 œ F œ a, showing, by 49, that 3 œ!, thereby establishing 18. Remark 4 We observed in Remark 3 that a sequence of iid copies of a random vector with dependent but uncorrelated coordinates satisfies the collection of assumptions 1, 2, 14, and 18. Clearly, if we have a sequence of iid copies of a random vector with coordinates that have a non-zero correlation coefficient, then 18 is violated and neither 9 nor 19 can hold. In this background, Proposition 1 is a generalization of the iid CLT for a sequence of random vectors. // Proposition 1 Assume 12. For all 4, let!á3 œ3 44. Then, for every subnet / 9 À Y such that / 9 œ,

MAJUMDAR AND MAJUMDAR 10 L œ F F 1 9 3 and, with 3 is as in 49, J œ a. 2 9 3 Proof of Proposition 1 Since 12 implies 14, and 24 is satisfied by every subsequence for the same 3, we conclude from Lemma 1 that L œ F F. 3 0 3 By 3 and 47, < 0=ß > œ < 3=ß >. 4 Recall that the decomposition of < into the three terms in 28 was obtained solely on the basis of 14 and, and on the basis of 1 and 2. Since 12 implies 1, 2, and 14, the decomposition of obtained in 28 continues to hold even though 18 is not true any more. With 4 substituting for 27, 31 with < on the RHS replaced by < 3 holds. Since the it, of every subnet / 9: À of / 9 À Y equals by Lemma A.1, RHS34 reduces to. Since 11 was obtained only by using 1 and 2, < 0 À Á is still uniformly equicontinuous; as such, RHS3, via 4, equals < 3=ß >, implying that 32 with < on the RHS replaced by < 3 holds as well. The same assertion is true for 36. Since 31, 32, and 36, with < replaced by < 3, continue to hold, we obtain < 9 =ß > œ < 3=ß >, implying 1 by 11 and the LCT for nets. Since, œ, by @ 3 œ Î 9 3 39 ; the proof of 2 follows as in the proof of 19 Ê 18 above. Proof of Lemma 1 Compactness of ß implies 24. By the LCT, 2 follows once 0 3 L Ø ß =ß> Ù œ Ø ß =ß> Ù =ß> d 7 <7 F F for every is proved by considering the cases 7=ß > œ! and 7=ß >! separately, where 3 a7 =ß> 7=ß > œ = => 3 > so that F F Ø ß =ß > Ù œ, 6 with 3 as in 24. By 14, Var =ß > ß ] œ = => >, implying 0 3 0 7=ß > œ Var =ß> ß] 7=ß > œ! =ß> ß] 7 0<7. If, 0<7 converges in quadratic mean, hence in distribution, to!, that is, 0 3 L Ø ß =ß> Ù œ!œ œ Ø ß =ß> Ù 7 <7 point mass at a! F F,

11 ASYMPTOTIC NORMALITY where the last equality follows from 6. If 7 =ß >!, we will use the subsequential Lindeberg CLT of Lemma A.6 to prove. In the notation of Lemma A.6, with we have =\ ß4. >\ ß4. [ 4 œ, =\ ß4. >\ ß4. [ œ œ =ß> ß] 4œ œ= => 344 >, and 7 œ= => 3 0 >. Since 7<7 œ 7 =ß>!, to 7 apply Lemma A.6 it remains to verify that 4 where, with the dependence on =ß > suppressed, 0 P œ!! 7 <7 % for every %, 7 =\ >\ =\ >\ ß4. ß4. ß4. ß4. P % œ E % 7 4œ 7. Recall from Lemma A.6 that P %, is well defined beyond a finite stage. Define =\ >\ =\ >\ ß4. ß4. ß4. ß4. % ß =ß> œ E %. 4œ Note that is decreasing in the first argument % for Á!, % ß =ß> œ =ß>. l l ß 8 There exists Q Á such that 7 Q implies 7<7 7=ß> Î. Consequently, by the definitions of P and, and using 8, for all 7 Q, 7 =ß> P ß <7 % <7 =ß> =ß > % 7. To establish 7 from here, it suffices to show that % ß =ß> œ! for every =ß> d and %!. 9 By the identical distribution and square integrability components of 1 and 2,

MAJUMDAR AND MAJUMDAR 12 % ß =ß> œ! for every %! 60 holds if =ß > equals either ß! or!ß. Therefore, 9 follows from 60 once we show that Pœ =ß> d À60 holds is a subspace. Clearly, P contains!ß! and is closed under scalar multiplication by 8. To verify that P is closed under vector addition, it suffices to show that for =ß > ß?ß @ d, % % % ß =ß>?ß@ Ÿ% ß =ß> % ß?ß@. 61 Adapting an argument from the proof of Theorem 1.3 of Kundu, Majumdar, and Mukherjee (2000), since \ ß4. \ ß4. =? >@ \ ß4. \ ß4. \ ß4. \ ß4. Ÿ max = > ß? @, \ ß4. \ ß4. =? >@ \ ß4. \ ß4. \ ß4. \ ß4. Ÿ% max= > ß? @ \ß4. œ% = > \ ß4 max. \ ß4 \ß4 ß?. @., and maxbß C maxbß C % Ÿ B B % C C %, 61 follows. By Lemma A.6, as 7Ä, <7 [ œ =ß>ß] converges in distribution to a 7=ß>, and follows from 6. <7 0<7 Proof of Lemma 2 By Lemma A.2, given an arbitrary subnet 3 Y 9 À 3 À Á Á, it suffices to find a further subnet 3 À such that 9 : 3 9: œ!. 62 Recall from 23 the existence of the directed set such that the subnet / 9: À converges to,. We will show that 62 holds for by separately considering the cases,!ß and,!ß. By the equivalence of (i) and (iv) in the Portmanteau Theorem [van der Vaart and Wellner (1996, Theorem 1.3.4)], inf I [ s and inf IY s Þ 63 of

13 ASYMPTOTIC NORMALITY Since, by 44, / / I [ s œ IYs œ / 3 and / 3, we obtain from 63 / / 3 œ!, implying 62 for,!ß. Since 3 Ÿ for all 4 implies 3 Ÿ 8 Î7, 44 LHS62 Ÿ 89: 79: ; 64 since, œ! implies RHS64 equals the it of / 9:, whereas, œ implies RHS64 equals the it of / 9:, 62 follows for,!ß. Remark Theorem 1 stands on the Lindeberg CLT for independent random vectors. It is worth investigating if we can obtain Theorem 1 when 14 is relaxed to 7-dependence or martingale difference array by using the CLTs under these dependence structures. // Remark 6 We conjecture that the joint distribution of the standardized sample means from random samples will be asymptotically Normal in d if an extension of 14 and an appropriate extension of 18 for pairwise correlation coefficients hold. // A.1. Nets and subnets. A set endowed with a reflexive, anti symmetric, and transitive binary relation is called a partially ordered set. The pair ß is called a directed set if, for each ß, there exists ( such that ( and (. Given a metric space Wß. and a directed set ß, a W valued net is defined to be a function BÀ ÈW; we write the net as B À. Recall that the net B À converges to B W if, for every %!, there exists! % such that! % implies.bßb %. Note that a W valued sequence is a W valued net indexed by Á. Let ß and ß be directed sets. Let 9 À È be order preserving, that is, 3 4Ê9 3 9, 4 and cofinal, that is, for each, there exists such that 9. Then the composite function C œ B 9, where B À È W, defines a net C À in W, is called a subnet of B À, and is written as B À. 9 Lemma A.1 Let be a directed set and B À a net taking values in W that converges to B W. Then every subnet of B À converges to B. Proof of Lemma A.1 This is Exercise 8 of page 188 of Munkres (2000). The proof follows from the definition of a subnet.

MAJUMDAR AND MAJUMDAR 14 Lemma A.2 Let ß be a directed set and B À a net taking values in W. Then B À converges to B W if and only if every subnet of B À has a further subnet that converges to B. Proof of Lemma A.2 Our inability to find a published proof of this very well-known result prompts us to sketch one here. The only if assertion follows from Lemma A.1. Conversely, suppose that every subnet of B À has a further subnet that converges to B. To prove by contradiction that B À converges to B, assume that B À does not converge to B, that is, there exists an %! such that for every w w, there exists satisfying with. B w ßB %. Note that w œ À. B w ßB % is a partially ordered set with the inherited relation. w w w Since ß is a directed set, given ß there exists ( such that ( w w w and (. But given (, there exists ( such that ( ( with. B( w ßB %; w w w w w that is, (. Since is transitive, ( and (, establishing that ß is a directed set. Clearly, the inclusion map from to is order preserving. The argument used to establish that ß is a directed set also establishes the cofinality of the w inclusion map. Consequently B w À is a subnet of B À. By the if assumption, B w À w has a subnet that converges to B, which is impossible since every element of B w À w is outside an % -neighborhood of B. Lemma A.3 W is compact if and only if every net in W has a convergent subnet. Proof of Lemma A.3 This is the theorem stated in Exercise 10 of page 188 of Munkres (2000), who sketches an outline of the proof as a hint. AÞ2 Þ Miscellaneous results from probability. We have used Lemmas A. and A.6 in the paper; Lemma A.4 is used in the proofs of Lemmas A. and A.6. Lemma A.4 For every ) dand 7!, 7 ; 7 u) ) exp u) Ÿ. 6 ;x 7 x ;œ! Proof of Lemma A.4 Again, the result is extremely well-known [Fabian and Hannan (198, page 14)], but we could not locate a reference with a proof, hence the sketch here. The inequality in 6 is vacuously true for ) œ!. Since expu) and expu), ; ; and, for every ;, u and u, are complex conjugates of each other, it suffices to prove 6 for )!. With - denoting the cosine function and = the sine function, 7 ; 7 ; 7 ; expu u ) ) ; ) œ - -! u = ; ) ) ) =! ;x ;x ;x ; that is because for 7!, ;œ! ;œ! ;œ!

1 ASYMPTOTIC NORMALITY - ; œ %7 if = if ; œ %7 ; = if ;œ%7 ; - if ;œ%7 - œ and = œ - if ;œ%7 = if ;œ%7 = if ;œ%7 - if ;œ%7. By Theorem 7.6 of Apostol (1967), with 1 equal to either - or =, 7 ; ) 1 ; 7 7 ) 1! œ ) > 1 >.>. ;x 7x ;œ! Thus, substituting?œ) >, ) ) 7 7 7 7 LHS6 œ? - )?.?? = )?.?. 7x!!! ) Consider the probability distribution. on!ß ) with Lebesgue density? 7 Î) ; then 6 follows from the variance inequality with respect to. and the fact that for every! and every B d, - B = B œ. Î 7 7 Lemma A. Let / 1 À1 Z ` d be uniformly tight. Let < 1 be the CF of / 1. Then < 1 À1 Z is a uniformly equicontinuous family of functions. Proof of Lemma A. Fix %! arbitrarily. Let Obe a compact subset of d such that - % / 1O for all 1 Z. 66 % By Lemma A.4 (with 7œ! ) and Cauchy-Schwartz inequality, for all =ß>ßB d exp u >=ßB Ÿ >=ßB Ÿ >= B. There exists œ % ÎQ! such that > = implies % supexpu >=ßB ÀB O, 67 where Q bounds the compact set O. Therefore, by 66 and 67, => implies % - < 1 = < 1 > / 1O / 1O %, completing the proof of the lemma. Lemma A.6 ( A subsequential Lindeberg CLT ) Let [ 4 À 4 Á be a sequence of independent random variables with E[ œ! and Var[ œ. For Á, let 4 4 4

MAJUMDAR AND MAJUMDAR 16 [ œ [ 4 and 7 œ 4 œ Var[. 4œ 4œ Let 7< À< Á be a subsequence of 7 À Á such that 7 œ 7! < <. 68 Then, there exists O Á such that 7! for all O. 69 For %! and O, let P % œ E 7 [ 7 [ % 4 4 4œ. Assume P œ!! < < % for every %. 70 Then, as <Ä, <[ converges in distribution to. 71 < a 7 Proof of Lemma A.6 First note that 68 rules out the possibility that 7 œ!a Á. Thus, there exists O Á such that 7O!. Since 7 À Á is a nondecreasing sequence, 69 follows. Thus we can, without loss of generality, assume that O œ. For every Á and Ÿ4Ÿ, let ^ ÀŸ4Ÿ be iid µ F; further, let ß4 ß4 4 ß4 ß4 ß4 4œ ] œ 7 [, 9 œ the CF of ], W œ ], and E œ the CF of W : œ Var] œ E ] œ 7 so that : œ ß4 ß4 ß4 4 ß4 4œ 72 ] s œ : ^, 9s œ the CF of ] s, Ws œ ] s, and Es œ the CF of Ws. ß4 ß4 ß4 ß4 ß4 ß4 4œ Clearly, for every > d, E > œ 9ß4 > 9 s, ß4 > œ exp > : ß4, and Es > œ exp >. 4œ Fix!Á> darbitrarily. We will show that

17 ASYMPTOTIC NORMALITY < < < E > > œ! œ s < < <ß4 > > < E < < <ß4, 73 4œ 4œ where which implies < < ß4 > œ > : ß4, 74 > œ exp >. 7 E < Note that 7 implies, by the LCT, Proposition 9.3.4 of Dudley (1989), and Lemma A., that E< À< Áis uniformly equicontinuous, whence, by 68, E > 7 œ exp > 7 < < <, implying, since [ œ 7 W, 71 by the LCT. Thus, it remains to show 73. Since ] ß4 À Ÿ 4 Ÿ is a row in the sense of Definition 4.1.4 of Fabian and Hannan (198), for arbitrary %!, by Lemma 4.1.7 of Fabian and Hannan (198), ß4 + œmax: ÀŸ4Ÿ Ÿ% P %, 76 implying, by 70, + œ! < <. 77 ß4 ß4 4œ By 72 and 76, Ps œ ] s ] s % % E % Ÿ I^ ^ ; by 77, s < P œ!! < % for every %. 78 With 2C œexp u>cu>c> C Î for C d, we obtain E2 ] ß4œ 9ß4 > < ß4 > E2] s œ 9 s > < >. ß4 By 6 with, ; with, 7œ 2 C Ÿ>C 7œ 2 C Ÿ > CC. For every %!, ß4 ß4 ß4 ß4 ß4 4œ 4œ 9 > < > Ÿ % > : > E ] ] % œ % > > P % ; similarly, ß4 ß4 +

MAJUMDAR AND MAJUMDAR 18 9 s > < > Ÿ % > > Ps %. 4œ ß4 ß4 By 77, there exists V œ V > Á such that < V implies, for all Ÿ 4 Ÿ <,!Ÿ: Ÿ %, <ß4 > so that by 74, for <V, > Ÿ. By Lemma 4.1.2 of Fabian and Hannan (198), for <V, < <ß4 < < <ß4 <ß4 ß4 ß4 < 4œ 4œ 4œ < < <ß4 <ß4 < ; 4œ 4œ 4œ 9 > < > Ÿ 9 > < > Ÿ % > > P % 9 s > < > Ÿ 9s ß4 > < ß4 > Ÿ % > > Ps % that establishes 73 by 70 and 78, completing the proof. References [1] Apostol, T. M. (1967). Calculus, Vol 1, Wiley, New York, NY. [2] Dudley, R. M. (1989). Real Analysis and Probability, Wadsworth & Brooks/Cole, Pacific Grove, CA. [3] Fabian, V. and Hannan, J. (198). Introduction to Probability and Mathematical Statistics, Wiley, New York, NY. [4] Kundu, S., Majumdar, S., and Mukherjee, K. (2000). Central Limit Theorems Revisited. Statist. Probab. Lett. 47 26-27. [] Munkres, J. R. (2000). Topology, Prentice Hall, Upper Saddle River, NJ. [6] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces, Academic Press, New York, NY. [7] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer, New York. RAJESHWARI MAJUMDAR SUMAN MAJUMDAR rajeshwari.majumdar@uconn.edu suman.majumdar@uconn.edu PO Box 47 1 University Place Coventry, CT 06238 Stamford, CT 06901