Central Limit Theorems revisited

Statistics & Probability Letters 47 (2000) 265–275

Central Limit Theorems revisited

Subrata Kundua ;∗, Suman Majumdarb, Kanchan Mukherjeec;1

aDepartment of Statistics, George Washington University, Washington, DC 20052, USAbUniversity of Connecticut, 1 University Place, Stamford, CT 06901-2315, USA

cDepartment of Statistics and Applied Probability, National University of Singapore, Singapore, 119260, Republic of Singapore

Received February 1999; received in revised form June 1999

Abstract

A Central Limit Theorem for a triangular array of row-wise independent Hilbert-valued random elements with �nitesecond moment is proved under mild convergence requirements on the covariances of the row sums and the Lindebergcondition along the evaluations at an orthonormal basis. A Central Limit Theorem for real-valued martingale di�erencearrays is obtained under the conditional Lindeberg condition when the row sums of conditional variances converge to a(possibly degenerate) constant. This result is then extended, �rst to multi-dimensions and next to Hilbert-valued elements,under appropriate convergence requirements on the conditional and unconditional covariances and the conditional Lindebergcondition along (orthonormal) basis evaluations. Extension to include Banach- (with a Schauder basis) valued randomelements is indicated. c© 2000 Elsevier Science B.V. All rights reserved

MSC: primary 60F05; secondary 60B12

Keywords: Central Limit Theorem; Hilbert space; Lindeberg condition; Martingale di�erence array; Weak convergence

1. Introduction and results

In this paper we prove four Central Limit Theorem (CLT)s. The �rst CLT (Theorem 1.1) is for a triangulararray of random elements taking values in a real separable in�nite dimensional Hilbert space, where thecomponents of the array are row-wise independent and have �nite second moment, the covariance operatorof the row sum satis�es mild convergence requirements, and the array satis�es the Lindeberg conditionalong the evaluation at an orthonormal basis. The second CLT (Theorem 1.2) is for a real-valued martingaledi�erence array for which the row sum of conditional variances converges in probability to a constant �2 anda conditional Lindeberg condition is satis�ed. The next one (Theorem 1.3) is an extension of Theorem 1.2 tomulti-dimensions and the last one (Theorem 1.4) is an extension of Theorem 1.3 to the Hilbert space setting.

∗ Corresponding author.1 The research is partly supported by the Academic Research Grant RP 3982708 of the National University of Singapore.

0167-7152/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reservedPII: S0167 -7152(99)00164 -9

266 S. Kundu et al. / Statistics & Probability Letters 47 (2000) 265–275

Remark 2.2 indicates how to extend Theorems 1.1 and 1.4 to a Banach space with a Schauder basis. Precisestatements follow.For a row-wise independent triangular array of real-valued random variables with �nite second moments,

the Lindeberg condition is necessary and su�cient (N–S) for the CLT if the array is uniformly asymptot-ically negligible (UAN), see Chung (1970, Theorem 7:2:1). Parthasarathy (1967, Theorem VI:6:3) obtainsN–S conditions for the CLT to hold for a triangular array of Hilbert-valued row-wise independent randomelements with �nite second moment, when the array is what he calls Uniformly In�nitesimal. Araujo andGin�e (1980, Chapter 3, Corollary 7:8) obtains su�cient conditions when the array is UAN and consistsof Type-p (16p62) Banach-valued row-wise independent random elements with square integrable norms.Garling (1976, Theorem 9) obtains su�cient conditions when the array consists of Type-2 Banach-valuedrow-wise independent random elements with square integrable norms. Even though Theorem 1.1 is subsumedby Garling’s result, its formulation will (hopefully) be more convenient for applications, as the assumptionsare relatively easier to verify and the limiting Gaussian distribution is easily identi�able from the simplesu�cient conditions. There is a vast literature on CLTs for dependent random elements, including martingaledi�erence arrays. See Hall and Heyde (1980), Helland (1982), Gaenssler and Haeusler (1986), Basu (1988)and Xie (1995), among others, for a review. However, we have not encountered any result that subsumes(or comes close to) Theorem 1.4, importance of which stems from the relatively easier to verify su�cientconditions and a simple proof.In the sequel, let H stand for a real separable in�nite dimensional Hilbert space; I{A} for the indicator

function of a set A; N(0; �2) for the normal distribution on the line with zero mean and variance �2; Nm(0; �)for the normal distribution on the m-dimensional Euclidian space with zero mean and covariance �; andG(0; C) for the normal distribution on H with zero mean and covariance operator C (as well as for aH -valued random element with the said distribution); i.e., for each h and g in H ,

〈Ch; g〉= E(〈G(0; C); h〉〈G(0; C); g〉):For an array {(Xn1; Xn2; : : : ; Xnn); n¿1} of real, multi-dimensional, or Hilbert-valued random elements, letSn =

∑nj=1 Xnj. Following Pollard (1984, Chapter VIII) we assume, without loss of generality, that there are

exactly n variables in the nth row.The following is an extension of Theorem 4:3:2 of Fabian and Hannan (1985) for row-wise independent

random vectors satisfying the Lindeberg condition.

Theorem 1.1. Let {ek : k¿1} be an orthonormal basis of H . For each positive integer n, let Xn1; Xn2; : : : ; Xnnbe a �nite sequence of independent H -valued random elements such that E(〈Xnj; el〉)=0 and E(‖Xnj‖2)¡∞for every 16j6n; l¿1. Let Cn be the covariance operator of Sn. Assume that the following conditionshold.(i) limn→∞ 〈Cnek ; el〉= akl, for all k¿1 and l¿1.(ii) limn→∞

∑∞k=1〈Cnek ; ek〉=

∑∞k=1 akk ¡∞.

(iii) limn→∞ Ln(�; ek) = 0 for every �¿ 0 and every k¿1, where for b ∈ H ,

Ln(�; b) :=n∑j=1

E(〈Xnj; b〉2I{|〈Xnj; b〉|¿�}):

Then

SnL→G(0; C); (1.1)

where the covariance operator C is characterized by 〈Ch; el〉=∑∞

j=1〈h; ej〉ajl, for every l¿1.

Note that for di�erent n the random elements can be de�ned on di�erent probability spaces.

S. Kundu et al. / Statistics & Probability Letters 47 (2000) 265–275 267

We now turn our attention to martingale di�erence arrays. For each n¿1, let Xn1; : : : ; Xnn be a �nite sequenceof real-, Rm- or H -valued random elements. Also, let Fn0⊆Fn1⊆ · · ·⊆Fnn be a �ltration in the probabilityspace underlying Xn1; : : : ; Xnn. Then {Xnj} is called a martingale di�erence array with respect to {Fnj} if foreach 16j6n and n¿1,(i) Xnj is Fnj measurable,and(ii) E(Xnj|Fn; j−1) = 0.Here the conditional expectation is de�ned coordinatewise for random vectors and in the Pettis sense forH -valued random elements. Since integrability of the squared norm is going to be a part of the hypotheses,the conditional expectation in the Pettis sense is going to be well de�ned by the Riesz–Frechet theorem. Asin the row-wise independent case, for di�erent n the random elements can be de�ned on di�erent probabilityspaces.The following CLT for real-valued martingale di�erence array is Theorem VIII:1 of Pollard (1984) strength-

ened to include �2 = 0.

Theorem 1.2. Let {Xnj} be a martingale di�erence array with respect to {Fnj} such that E(X 2nj)¡∞ forevery 16j6n; n¿1. Assume that the following conditions hold.(i)∑n

j=1 E(X2nj|Fn; j−1)

Pr−→ �2, for some �¿0.

(ii)∑n

j=1 E(X2njI{|Xnj|¿�}|Fn; j−1)

Pr−→ 0, for every �¿ 0.Then

SnL→N(0; �2): (1.2)

Note that the second condition is the Lindeberg condition stated in terms of the conditional expectation.Next we extend Theorem 1.2 when the martingale di�erence array is Rm-valued.

Theorem 1.3. Let {Xnj} be a Rm-valued martingale di�erence array with respect to {Fnj} such thatE(‖Xnj‖2)¡∞ for every 16j6n; n¿1. Let {e1; : : : ; em} be any basis of Rm. Assume that the followingconditions hold.(i) For every b ∈ Rm; ∑n

j=1 E((b′Xnj)2|Fn; j−1)

Pr−→ �2b for some �b.

(ii) Ln(�; ek)Pr−→ 0 for every �¿ 0 and every 16k6m, where for b ∈ Rm

Ln(�; b) :=n∑j=1

E[(b′Xnj)2I{|b′Xnj|¿�}|Fn; j−1]:

Then

SnL→Nm(0; �); (1.3)

where the covariance matrix � is characterized by b′�b= �2b, for all b ∈ Rm.

Finally, we extend Theorem 1.3 when the martingale di�erence array is H -valued.

Theorem 1.4. Let {Xnj} be a H -valued martingale di�erence array with respect to {Fnj} such that E(‖Xnj‖2)¡∞ for each 16j6n; n¿1. Let {ek : k¿1} be an orthonormal basis of H . Assume that the followingconditions hold.(i) For every b ∈ H; ∑n

j=1 E(〈Xnj; b〉2|Fn; j−1)Pr−→ �2b for some �b.

(ii) limn→∞∑∞

k=1

∑nj=1 E(〈Xnj; ek〉2) =

∑∞k=1 �

2ek ¡∞.


(iii) Ln(�; ek)Pr−→ 0 for every �¿ 0 and every k¿1, where for b ∈ H ,

Ln(�; b) :=n∑j=1

E(〈Xnj; b〉2I{|〈Xnj; b〉|¿�}|Fn; j−1):

Then

SnL→G(0; C); (1.4)

where the covariance operator C is characterized by 〈Cb; b〉= �2b, for all b ∈ H .

2. Proofs and extensions

We �rst present the technique that will be used to extend the results from �nite to in�nite dimensions,namely, Convergence in Distribution via Uniform Approximation. See Pollard (1984, Example IV:11) andAraujo and Gine (1980, p. 34) for more on it.

Proposition 2.1. Let H be a real separable Hilbert space with orthonormal basis {ek : k¿1}. For N¿1,let �N denote the orthogonal projection onto the linear span of {e1; : : : ; eN}. Let X; X1; X2; : : : be H -valuedrandom elements satisfying the following two conditions.

For every N¿1; as n→ ∞; �N (Xn)L→�N (X ): (2.1)

limN→∞

lim supn→∞

Pr[‖Xn −�N (Xn)‖¿�] = 0: (2.2)

Then

XnL→X:

Proof. For a bounded and continuous function f : H 7→ R,

|Ef(Xn)− Ef(X )|6 |Ef(Xn)− Ef(�N (Xn))|+ |Ef(�N (Xn))− Ef(�N (X ))|+|Ef(�N (X ))− Ef(X )|: (2.3)

The iterated limit, limN→∞ lim supn→∞, of the �rst term in the right-hand side is arbitrarily small, by thecontinuity and boundedness of f plus the second assumption (2.2), applied via the split of the probabilityspace underlying Xn into [‖Xn −�N (Xn)‖¿�] and its complement. The second term goes to zero (for �xedN , as n→ ∞) by the �rst assumption (2.1). The third term in the right-hand side goes to zero (as N → ∞)by the de�nition of �N and the continuity and boundedness of f.

For the sake of completeness and ease of reference, we quote below (using our notation and terminology)Theorem 4:3:2 of Fabian and Hannan (1985).

Theorem FH. For each positive integer n; let (Yn1; : : : ; Ynn) be a �nite sequence of independent m-dimensionalrandom vectors with zero expectations and sum Tn :=

∑nj=1 Ynj. Let � be a matrix such that limn→∞ E(TnT ′

n)= �. Let B be a subset of Rm such that every column of � is a linear combination of elements in B. Forevery b in B and �¿ 0; assume that

limn→∞

n∑j=1

E((b′Ynj)2I{|b′Ynj|¿�}) = 0:


Then

TnL→Nm(0; �):

Proof of Theorem 1.1. We �rst show that C is well de�ned. The de�nition of Cn, the Cauchy–Schwartzinequality and assumption (i) imply that ajl6a

1=2jj a

1=2ll . Consequently, for each l¿1,

〈Ch; el〉26all( ∞∑

j=1

〈h; ej〉a1=2jj)26all

( ∞∑j=1

〈h; ej〉2)( ∞∑

j=1

ajj

):

Therefore,∑∞

l=1〈Ch; el〉26‖h‖2(∑∞k=1 akk)

2. Assumption (ii) now shows that C is a trace class operator,guaranteeing the existence of G(0; C).Now we use Theorem FH to verify (2.1). We need to show that for each N¿1,

(〈Sn; e1〉; : : : ; 〈Sn; eN 〉)′ L→N (0; A);

where A := [akl]. To apply Theorem FH, set

Ynj := (〈Xnj; e1〉; : : : ; 〈Xnj; eN 〉)′ ∀16j6n; n¿1:Then Tn = (〈Sn; e1〉; : : : ; 〈Sn; eN 〉)′ and by assumption (i),

E(TnT ′n) = [〈Cnek ; el〉]→ � = A:

With B equal to the collection of the columns of the N -dimensional identity matrix, (2.1) follows by TheoremFH and assumption (iii).To verify (2.2), note that by Chebychev’s inequality,

Pr[‖Sn −�N (Sn)‖¿�]6 E‖Sn −�N (Sn)‖2=�2

=∞∑

k=N+1

〈Cnek ; ek〉=�2:

Therefore, by assumptions (i), (ii) and Lemma A.1 (applied with the counting measure),

lim supn→∞

Pr[‖Sn −�N (Sn)‖¿�]6∞∑

k=N+1

akk=�2:

Since∑∞

k=1 akk ¡∞, the right-hand side goes to zero as N → ∞.

Proof of Theorem 1.2. When �¿ 0, this is Theorem VIII:1 of Pollard (1984). Hence, we consider thecase � = 0 only. We show that if {Xnj; 16j6n} is a martingale-di�erence array with conditional variancevnj := E[X 2nj|Fnj−1] such that

∑nj=1 vnj

Pr−→ 0, then SnPr−→ 0.

De�ne a sequence of stopping time �n = sup{k6n; ∑kj=1 vnj61}. Since

∑nj=1 vnj

Pr−→ 0, we getlimn→∞ Pr[�n¿n] = 1. Hence

limn→∞Pr

[n∑j=1

Xnj 6=n∑j=1

XnjI{j6�n}]6 lim

n→∞Pr[�n¡n] = 0:

Therefore, it is enough to show that∑n

j=1 XnjI{j6�n} Pr−→ 0, where by the Optimal Sampling Theorem,

{XnjI{j6�n}; 16j6n} is again a martingale di�erence array. In the sequel, let Wk denote∑k

j=1 XnjI{j6�n}and Ek denote the conditional expectation with respect to the kth sigma-�eld Fn;k , where the dependence on n


of some of the quantities is suppressed for notational simplicity. Writing Wk =Wk−1 +XnkI{k6�n}, squaringboth sides and taking conditional expectation with respect to the (k − 1)th sigma-�eld, we obtain that for26k6n,

Ek−1[W 2k ] =W

2k−1 + vkI{k6�n}:

Hence,

E1W 22 + E2W

23 + · · ·+ En−1W 2

n

=W 21 +W

22 +W

23 + · · ·+W 2

n−1 + v2I{26�n}+ · · ·+ vnI{n6�n}:Now, by taking expectation on both sides and cancelling EW 2

2 ; : : : ; EW2n−1, we get

EW 2n = E[W

21 + v2I{26�n}+ · · ·+ vnI{n6�n}]

= E[v1I{16�n}+ v2I{26�n}+ · · ·+ vnI{n6�n}]:Note that by the de�nition of �n,

v1I{16�n}+ v2I{26�n}+ · · ·+ vnI{n6�n}61:Also, by assumption

v1I{16�n}+ v2I{26�n}+ · · ·+ vnI{n6�n} Pr−→ 0:

Hence limn→∞ EW 2n = 0, completing the proof.

Proof of Theorem 1.3. Since convergence in Rm is determined by convergence of linear functionals, it isenough to show that b′Sn

L→N(0; b′�b) for every b ∈ Rm. By Theorem 1.2, we need only to show thatLn(�; b)

Pr−→ 0 for every �¿ 0 and b ∈ Rm. Following the proof of Theorem 4:3:2 of Fabian and Hannan(1985), we de�ne L= {b ∈ Rm; Ln(�; b)= op(1) for every �¿ 0} and show that L is a linear subspace. Sinceby assumption (ii) L contains a basis, this completes the proof.

Clearly, 0 ∈ L. For � 6= 0; Ln(�; �b) = �2Ln(�=|�|; b), and consequently �b ∈ L if b ∈ L. Using|(b+ d)′Xnj|6|b′Xnj|+ |d′Xnj|6max{2|b′Xnj|; 2|d′Xnj|}

and

max(x; y)2I{max(x; y)¿�}6x2I{x¿ �}+ y2I{y¿�};we get

Ln(�; b+ d)64Ln(�=2; b) + 4Ln(�=2; d):

Hence b+ d ∈ L if b ∈ L and d ∈ L, completing the proof that L is a subspace.

Proof of Theorem 1.4. First we show that C is a well-de�ned trace class operator to ensure the existence ofG(0; C). Using xy = {(x + y)2 − x2 − y2}=2, and assumptions (i) and (ii), we get along a subsequence

n∑j=1

E(〈Xnj; h〉〈Xnj; g〉|Fn; j−1)a:s→ 〈Ch; g〉:

Hence by the Cauchy–Schwartz inequality,

〈Ch; g〉6〈Ch; h〉1=2〈Cg; g〉1=2:Now the desired conclusion follows as in the proof of Theorem 1.1.


Assumptions (i) and (iii), in conjunction with Theorem 1.3, imply that for every N¿1, as n→ ∞,(〈Sn; e1〉; : : : ; 〈Sn; eN 〉)′ L→N (0; A);

where A := [aij] is a N × N matrix and 2aij = �2ei+ej − �2ei − �2ej . Thus, (2.1) is veri�ed.To verify (2.2), observe that

E〈Sn; ek〉2 =n∑j=1

E〈Xnj; ek〉2 + 2n∑r=1

n∑j=r+1

E|〈Xnj; ek〉〈Xnr; ek〉|

=n∑j=1

E〈Xnj; ek〉2; (2.4)

since

E[{〈Xnj; ek〉〈Xnr; ek〉}|Fnr] = 0 for all r ¡ j:

Also, by Fatou’s lemma and assumption (i), using a subsequential argument,

lim infn→∞

n∑j=1

E(〈Xnj; ek〉2)¿〈Cek ; ek〉:

Hence, assumption (ii) and Lemma A.1 (applied with the counting measure) imply that

limn→∞

∞∑k=1

∣∣∣∣∣∣n∑j=1

E〈Xnj; ek〉2 − 〈Cek ; ek〉∣∣∣∣∣∣= 0: (2.5)

Next, applying Chebychev’s inequality, (2.4) and the Monotone Convergence Theorem, followed by (2.5),we obtain

Pr[‖Sn −�NSn‖¿�] 6 E

[ ∞∑k=N+1

〈Sn; ek〉2]/

�2

=∞∑

k=N+1

n∑j=1

E〈Xnj; ek〉2/�2

n→∞→∞∑

k=N+1

〈Cek ; ek〉/�2:

Since by assumption (ii)∑∞

k=1〈Cek ; ek〉¡∞, (2.2) is veri�ed.

We end this section with two remarks. The �rst one is on the assumption of Theorem 1.4, while the secondone is on Banach space extension.

Remark 2.1. Assumption (i) of Theorem 1.4 is equivalent to the following assumption:For every k; l¿1

n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1)Pr−→ 〈Cek ; el〉: (2.6)

That (i) implies (2.6) is obvious. We are going to show that (2.6) implies a strengthened version of (i)where the convergence in probability is replaced by convergence in L1.


To start with, note that (2.6) implies

n∑j=1

E(〈Xnj; ek〉2|Fn; j−1)Pr−→ 〈Cek ; ek〉

which self-strengthens, by a standard subsequential argument using (2.5) and Lemma A.1 (applied with theunderlying probability measure), to

n∑j=1

E(〈Xnj; ek〉2|Fn; j−1)L1−→ 〈Cek ; ek〉: (2.7)

We then show thatn∑j=1

E(〈Xnj; b〉2|Fn; j−1)a:s:=

∞∑k=1

∞∑l=1

〈b; ek〉Y (n)kl 〈b; el〉; (2.8)

where

Y (n)kl =n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1): (2.9)

Since

〈Xnj; b〉2 = limN→∞

TN ;

where

TN =N∑k=1

N∑l=1

〈b; ek〉〈Xnj; ek〉〈Xnj; el〉〈b; el〉;

(2.8) involves interchangeability of limN→∞ and∑n

j=1 E(·|Fn; j−1). Clearly TN6‖b‖2‖Xn;j‖2, whence (2.8)follows by the Dominated Convergence Theorem.Clearly, assumption (i) follows if we can show that

limn→∞

∞∑k=1

∞∑l=1

|〈b; ek〉〈b; el〉|t(n)kl = 0; (2.10)

where

t(n)kl = E

∣∣∣∣∣n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1)− 〈Cek ; el〉∣∣∣∣∣ : (2.11)

At this stage, the idea is to show that t(n)kl converges to 0 as n→ ∞, and use Lemma A.2 to obtain (2.10).Since for �; �¿0,

(�1=2 − �1=2)26|�− �|; (2.12)

from (2.7)[n∑j=1

E(〈Xnj; ek〉2|Fn; j−1)

]1=2L2−→ 〈Cek ; ek〉1=2:


Since by the Cauchy–Schwartz inequality (applied twice)∣∣∣∣∣n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1)

∣∣∣∣∣6[

n∑j=1

E(〈Xnj; ek〉2|Fn; j−1)

]1=2 [ n∑j=1

E(〈Xnj; el〉2|Fn; j−1)

]1=2

(2.13)

by a subsequential argument, we obtain from (2.6), (2.13) and Lemma A.2,n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1)L1−→ 〈Cek ; el〉 (2.14)

showing the convergence of t(n)kl to 0. By the triangle and (four applications of) Cauchy–Schwartz inequalities

t(n)kl = E

∣∣∣∣∣n∑j=1

E(〈Xnj; ek〉〈Xnj; el〉|Fn; j−1)− 〈Cek ; el〉∣∣∣∣∣

6

(n∑j=1

E〈Xnj; ek〉2n∑j=1

E〈Xnj; el〉2)1=2

+ (〈Cek ; ek〉〈Cel; el〉)1=2

= p(n)kl say: (2.15)

Also, a triangulation, (2.12) and (2.5) give∞∑k=1

∞∑l=1

|〈b; ek〉〈b; el〉‖p(n)kl − 2〈Cek ; ek〉1=2〈Cel; el〉1=2| → 0;

whence by (2.15) and Lemma A.2 (applied with the product of counting measures), (2.10) obtains.

Remark 2.2. Proposition 2.1 establishes su�ciency of (2.1) and (2.2) for the convergence in distribution ofXn to X . Since the orthogonal projection is continuous, (2.1) is clearly necessary. Since the norm is continuous,

XnL→X implies, ∀N¿1, as n→ ∞,

‖Xn −�N (Xn)‖ L→‖X −�N (X )‖: (2.16)

By the Portmanteau Theorem (Parthasarathy, 1967, Theorem II.6.1(c)),

lim supn→∞

Pr[‖Xn −�N (Xn)‖¿�]6Pr[‖X −�N (X )‖¿�]

and hence (2.2) obtains. Thus, (2.1) and (2.2) are necessary as well.Consequently, it is possible to replace assumption (ii) of Theorem 1.1 (or, for that matter, Theorem 1.4)

by the weaker assumption (2:2) and∑akk ¡∞ (

∑�2ek ¡∞), even though we doubt if it will be feasible

to verify (2.2) when the convergence assumption in (ii) fails to hold. Nonetheless, that observation suggeststhe following extension of Theorems 1.1 and 1.4 to Banach spaces with Schauder bases.Let B be a (necessarily separable) Banach space with Schauder basis {ei: i¿1} (Dunford and Schwartz,

1958, p. 93). By Exercise II.4.9 of Dunford and Schwartz (1958), x :=∑∞

i=1 �iei�k→ �k is continuous. It is

clearly linear. Unique determination of the coe�cient of �i in the Schauder expansion implies that {�i: i¿1}is linearly independent in B∗.

Extension of Theorem 1.1. Let G(0; C) be a Gaussian measure on B with mean 0 and covariance C. Let{(Xn1; : : : ; Xnn): n¿1} be a triangular array of row-wise independent random elements taking values in B


such that E(�(Xnj))= 0 for every � ∈ B∗, and E‖Xnj‖2¡∞, for every 16j6n; n¿1. Let Sn :=∑n

j=1 Xnj.Assume that the following conditions hold.(i) limn→∞

∑nj=1 E�i(Xnj)�k(Xnj) = �k(C�i) ∀i; k.

(ii) limn→∞ Ln(�; �k) = 0 ∀k, where

Ln(�; �) :=n∑j=1

E[{�(Xnj)}2I{|�(Xnj)|¿�}]:

(iii) limN→∞ lim supn→∞ Pr[‖Sn −∑N

k=1 �k(Sn)ek‖¿�] = 0 for each �¿ 0.Then

SnL→G(0; C):

The �rst two conditions will imply that ∀N¿1,

(�1(Sn); : : : ; �N (Sn))L→NN (0; A);

where A= [aik ]; aik = �i(C�k) and the last condition will make the uniform approximation idea of Proposi-tion 2.1 work.

Extension of Theorem 1.4. Let {(Xn1; : : : ; Xnn): n¿1} be a martingale di�erence array taking values in Bwhere the conditional expectation is de�ned in the Pettis sense. Assume that the following conditions hold.(i)∑n

j=1 E[(�(Xnj))2|Fn; j−1]

Pr−→ �(C�) ∀� ∈ B∗.(ii) Ln(�; �k)

Pr→ 0 ∀k, where

Ln(�; �) :=n∑j=1

E[{�(Xnj)}2I{|�(Xnj)|¿�}|Fn; j−1]:

(iii) limN→∞ lim supn→∞ Pr[‖Sn −∑N

k=1 �k(Sn)ek‖¿�] = 0.Then

SnL→G(0; C):

We conclude this remark by observing that while there are separable Banach spaces without any Schauderbasis (En o, 1973), many familiar Banach spaces do admit one. See Dunford and Schwartz (1958, p. 94).

Appendix

Lemma A.1 (An extension of Sche�e’s Lemma). Let f be a measurable function which is integrable withrespect to a �-�nite measure �. If {fn} is a sequence of nonnegative measurable functions such thatlim inf n→∞ fn¿f a: e: (�) and

∫fn d� → ∫

f d�; then 06lim inf n→∞ fn = f a: e: (�) and limn→∞∫ |fn −

f| d� = 0.

Proof. By Fatou’s lemma and the second assumption,∫(lim inf n→∞ fn) d�6

∫f d�;


whence the �rst conclusion follows from the �rst assumption. For the second conclusion, since |fn − f| =2 max(fn; f)− fn − f, it is enough to show that

limn→∞

∫[max(fn; f)− fn] d� = 0:

Now, ∫[max(fn; f)− fn] d� =

∫|f − fn|I

{lim infk→∞

fk = f¿fn

}f d�;

where the integrand in the above is bounded by f. To show the a.e. (�) convergence of the integrand to 0,�x ! such that lim inf k→∞ fk(!)=f(!). For every �¿ 0; ∃n0(�) such that for n¿n0(�); f(!)−fn(!)¡�,whence 06[f(!) − fn(!)]I{lim inf k→∞ fk = f¿fn}(!)¡�. The asserted L1 convergence follows by theDominated Convergence Theorem.

Lemma A.2 (Extended Dominated Convergence Theorem). Let � be a �-�nite measure and {fn} be a se-quence of measurable functions in L1(�), converging a.e. (�) to f. If {gn} is a sequence of nonnega-tive measurable functions in L1(�) converging to g in L1(�) such that |fn|6gn; then f is in L1(�) andlimn→∞

∫ |fn − f| d� = 0.

Proof. Follows by applying the necessity of the conditions of Vitali Convergence Theorem (Dunford andSchwartz, 1958, Theorem III.6.15) to {gn}, and the su�ciency of them to {fn}.

References

Araujo, A., Gin�e, E., 1980. The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.Basu, A.K., 1988. Uniform and nonuniform estimates in the clt for Banach valued dependent random variables. J. Multivariate Anal. 25,153–163.

Chung, K.L., 1970. A Course in Probability Theory. Academic Press, New York.Dunford, N., Schwartz, J.T., 1958. Linear Operators. Part I: General Theory. Wiley Classics Library Edition 1988. Wiley, New York.En o, P., 1973. A counter-example to the approximation problem in Banach spaces. Acta Math. 103, 309–317.Fabian, V., Hannan, J., 1985. Introduction to Probability and Mathematical Statistics. Wiley, New York.Gaenssler, P., Haeusler, E., 1986. On martingale central limit theory. In: Taqqu, M., Eberlein, E. (Eds.), Dependence in Probability andStatistics; A Survey of Recent Results. Birkhauser, Boston.

Garling, D.J.H., 1976. Functional central limit theorems in Banach spaces. Ann. Probab. 4, 600–611.Hall, P., Heyde, C.C., 1980. Martingale Limit Theory and Its Applications. Academic Press, New York.Helland, I.S., 1982. Central limit theorems for martingales with discrete or continuous time. Scand. J. Statist. 9, 79–94.Parthasarathy, K.R., 1967. Probability Measures on Metric Spaces. Academic Press, New York.Pollard, D., 1984. Convergence of Stochastic Processes. Springer, New York.Xie, Y., 1995. Limit theorems of Hilbert valued semimartingales and Hilbert valued martingale measures. Stochastic Process. Appl. 59,277–295.

Central Limit Theorems revisited

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