ARTICLE IN PRESS

Statistics & Probability Letters 77 (2007) 952–963 www.elsevier.com/locate/stapro

On complete convergence of triangular arrays of independent random variables Istva´n Berkesa,,1, Michel Weberb a

Department of Statistics, Technical University Graz, Steyrergasse 17/IV, A-8010 Graz, Austria Mathe´matique (IRMA), Universite´ Louis-Pasteur et C.N.R.S., 7 rue Rene´ Descartes, 67084 Strasbourg Cedex, France

b

Received 23 March 2006; received in revised form 11 September 2006; accepted 19 December 2006 Available online 19 January 2007

Abstract Given P a triangular array a ¼ fan;k ; 1pkpkn ; nX1g of positive reals, we study the complete convergence property n of T n ¼ kk¼1 an;k X n;k for triangular arrays X ¼ fX n;k ; 1pkpkn ; nX1g of independent random variables. In the Gaussian case we obtain a simple characterization of density type. Using Skorohod representation and Gaussian randomization, we then derive sufficient criteria for the case when X n;k are in Lp , and establish a link between the Lp -case and L2p -case in terms of densities. We finally obtain a density type condition in the case of uniformly bounded random variables. r 2007 Elsevier B.V. All rights reserved. MSC: primary 60G50; 60F15; secondary 60G15 Keywords: Complete convergence; Triangular arrays; Independent random variables

1. Introduction and results Throughout this paper, we let X ¼ fX n;k ; 1pkpkn ; nX1g denote a triangular array of real centered independent random variables, and a ¼ fan;k ; 1pkpkn ; nX1g with fkn ; nX1g non-decreasing, a triangular array of positive reals. When the random variables are symmetric (resp. identically distributed), we will say that the triangular array X is symmetric (resp. iid). Set, for every nX1, Tn ¼

kn X

an;k X n;k ;

k¼1

An ¼

kn X k¼1

an;k ; B2n ¼

kn X

a2n;k ; C n ¼ An =Bn .

(1)

k¼1

Let ðO; A; PÞ be the basic probability space on which X is defined. Note that C n X1. We investigate under c:c: what conditions the sequence T n =An converges completely to 0: T n =An ! 0, which means, as is well-known, Corresponding author. 1

E-mail addresses: [email protected] (I. Berkes), [email protected] (M. Weber). Research supported by Hungarian National Foundation for Scientific Research, Grants T 043037 and K 61052.

0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.12.011

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that for any e40 X PfjT n j=An 4ego1. n

The study of this property originates from a well-known paper by Hsu andP Robbins (1947) who proved in the case of a single iid sequence n ¼ fx; xn ; nX1g with partial sums S n ¼ nk¼1 xk , n ¼ 1; 2; . . . that Ex ¼ 0, c:c: + (1949) proved the validity of the converse implication. Ex2 o1 imply Sn =n ! 0. Shortly afterward, Erdos Since then, the study of various possible generalizations of this result (subsequence case, the theorems of Baum and Katz (1965), extensions to triangular arrays of independent random variables, Banach space valued random variables) have received a lot of attention. See, for example, the works of Pruitt (1966), Rohatgi (1971), Fazekas (1985, 1992), Hu et al. (1989), Kuczmaszewska and Szynal (1988, 1990, 1994), Gut (1992), Li et al. (1992), Rao et al. (1993), Sung (1997), Adler et al. (1999), Hu et al. (1999), Ahmed et al. (2002). The purpose of the present paper is to present new necessary as well as sufficient criteria for the complete convergence of triangular arrays of independent random variables, and discuss their relations with known results in the literature. We start our investigations with the Gaussian case, because of the classical Gaussian randomization procedure for sums of independent random variables, and also because this case is in general very informative. If X is Gaussian, the problem can be simply settled. Put LðaÞ ¼ lim sup x!1

log ]fn : C n pxg . x2

Then we have the following characterization. Theorem 1. Assume that the X n;k are iid standard Gaussian variables. Then we have c:c:

T n =An ! 0 () LðaÞ ¼ 0. In view of this complete result, it is natural to attack the general iid case using invariance principles. Applying Skorohod embedding for the row sums of the triangular array X leads, under natural conditions on c:c: the stopping times in the Skorohod representation, to a necessary and sufficient criterion for T n =An ! 0, see Proposition 10. This condition, in turn, leads to sufficient criteria under the existence of higher moments. In particular, we will prove: Theorem 2. Assume that EX 2n;k ¼ 1 and X n;k 2 L2p for some pX2. Then the relation P n 4 p=2 X ð kk¼1 an;k Þ Pkn 2 p o1, ð k¼1 an;k Þ n c:c:

implies T n =An ! 0. To compare this result with the Gaussian case, note that LðaÞ ¼ 0 is equivalent to 0  2 1 Pk n a X n;k k¼1 B C exp@d Pkn Ao1 for all d40. 2 n k¼1 an;k In the case when X is also symmetric, the condition in Theorem 2 can be weakened. Theorem 3. Assume that X is symmetric, EX 2n;k ¼ 1 and X n;k 2 L2p for some pX2. Then the relation P n 4 p=2 X ð kk¼1 an;k Þ o1, Pk n 2p p n ð k¼1 an;k Þ log n c:c:

implies T n =An ! 0. Recall that the array X is stochastically bounded by a random variable X if there is a constant D such that PfjX n;k j4xgpDPfjDX j4xg for all x40 and for all nX1, 1pkpkn . We will prove the following result.

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Theorem 4. Let X be a symmetric triangular array stochastically bounded by a square integrable random variable X. Assume that for any e40: P

1pkplo1 PfjX jXeAl =ak go1. Further assume that for some integer rX2 and any e40, P (b) nX1 PfjT n j4eAn gr o1. Then c:c: (c) T n =An ! 0. Conversely, if the triangular array X is iid symmetric, then (c) implies (a).

(a)

The next result concerns the uniformly bounded case. We show that a condition similar to that assumed in the Gaussian case suffices for complete convergence. Put, for any positive integer n, V 2n ¼

kn X

a2n;k X 2n;k .

k¼1

Theorem 5. Let X be a triangular array of real centered, uniformly bounded independent random variables. Assume that for any e40 E sup mX1

]fn : moAn =V n pm þ 1g o1. expfem2 g c:c:

Then T n =An ! 0. Our final result establishes a link between the complete convergence of arrays in the Lp and L2p -case. Remarkably, the link is provided by the density condition in the Gaussian case in Theorem 1. We need a preliminary definition. Definition. Let pX2. We say that a is p-regular if any triangular array X of real centered iid random variables c:c:

with finite pth moments satisfies T n =An ! 0. Let a be a triangular array of positive reals. Define a2 :¼fa2n;k ; 1pkpkn ; nX1g. Then we have: Theorem 6. Let pX2 and assume that a2 is p-regular. Then a is 2p-regular iff LðaÞ ¼ 0. 2. Proofs Proof of Theorem 1. Before giving the proof, recall for the reader’s convenience an elementary estimate for Gaussian random variables due to Komatsu–Pollak (see Mitrinovic´, 1970, p. 178). R1 2 2 Lemma 7. The Mills’s ratio RðxÞ ¼ ex =2 x et =2 dt satisfies rffiffiffi 2 2 p ffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi q for all x40. pRðxÞp p 2 2 8 x þ4þx x2 þ p þ x

Note that PfjT n j=An 4eg ¼ PfjNð0; 1Þj4eC n g 

2 1 eðeC n Þ =2 1 þ eC n

as n ! 1,

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where the symbol  means that the ratio of the two sides is between positive constants. Thus it follows that c:c: T n =An ! 0 if and only if the series X 2 edC n n

converges for any d40. And this is equivalent to LðaÞ ¼ 0 (for a proof, see e.g. Weber, 1995, pp. 402–403). & Proof of Theorem 6. The proof relies upon several intermediate results. Let n ¼ fxk ; kX1g be a sequence of real centered independent square integrable random variables defined on the probability space ðO; A; PÞ, and let w ¼ fwk ; kX1g be a sequence of positive reals. Put, for any positive integer m, Sm ¼

m X

w k xk ;

Wm ¼

k¼1

m X k¼1

wk ;

Mm ¼

m X

w2k .

k¼1

Recall the Skorohod embedding scheme (see e.g. Breiman, 1968): there exists, after suitably enlarging the probability space, a linear Brownian motion B ¼ fBðtÞ; 0pto1g starting at 0, and a sequence t1 ; t2 ; . . . of independent non-negative random variables with Etk ¼ w2k Ex2k , kX1 such that, with t0 ¼ 0 a.s., ( ! ! ) k k1 X X D B tj  B tj ; kX1 ¼fwk xk ; kX1g. j¼0

j¼0

Put, for any real x, Z 1 1 2 CðxÞ ¼ pffiffiffiffiffiffi eu =2 du. 2p x

pffiffiffiffiffi Lemma 8. Let e; h; d be positive numbers with e4h4 2d, and put  ( )  X m   Dm ¼ Dm ðdÞ ¼ P  t  M m XdM m .   j¼0 j Then, for any positive integer m, we have         Wm hW m Wm hW m C ðe þ hÞ pffiffiffiffiffiffiffiffi  4C pffiffiffiffiffiffiffiffiffiffiffiffiffi  Dm pPfjS m j4eW m gpC ðe  hÞ pffiffiffiffiffiffiffiffi þ 4C pffiffiffiffiffiffiffiffiffiffiffiffiffi þ Dm . Mm Mm 2dM m 2dM m Proof. We observe that ( ! )   X m   PfjS m j4eW m g ¼ P B tj 4eW m   j¼0

( ! )   X m   tj 4eW m ; jBðM m Þjpðe  hÞW m pPfjBðM m Þj4ðe  hÞW m g þ P B   j¼0  ( ! )     X m ðe  hÞW m   pffiffiffiffiffiffiffiffi tj  BðM m ÞXhW m pC þ P B   Mm j¼0   ( ) ( )    X m ðe  hÞW m   pffiffiffiffiffiffiffiffi t  M m XdM m þ P sup jBðyM m Þ  BðM m ÞjXhW m pC þP    j¼0 j Mm jy1jpd   ( ) ( )     m ðe  hÞW m Wm  X pffiffiffiffiffiffiffiffi ¼C t  M m XdM m þ P sup jBðyÞ  Bð1ÞjXh pffiffiffiffiffiffiffiffi . þP    j¼0 j Mm Mm jy1jpd

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Conversely,   Wm C ðe þ hÞ pffiffiffiffiffiffiffiffi ¼ PfjBðM m Þj4ðe þ hÞW m g Mm  ( ! ) ( ! )    X  X m m     p P B tj 4eW m þ P jBðM m Þj4ðe þ hÞW m ; B tj peW m     j¼0 j¼0  ( ! ) ( ! )    X  X m m     p P B tj 4eW m þ P B tj  BðM m ÞXhW m     j¼0 j¼0  ( ! ) ( )   X  X m m     p P B tj 4eW m þ P  tj  M m XdM m     j¼0 j¼0 ( ) þP

sup jBðyM m Þ  BðM m ÞjXhW m jy1jpd

 ( ! ) ( ) ( )   X  X m m Wm     ¼ P B tj 4eW m þ P  t  M m XdM m þ P sup jBðyÞ  Bð1ÞjXh pffiffiffiffiffiffiffiffi .    j¼0 j  Mm jy1jpd j¼0

Since B has stationary increments, we get by using scale invariance, the symmetry of the law of B and Eq. (1.5.1) in Cso¨rgo+ and Re´ve´sz (1981, p. 43), ( ) ( ) Wm hW m P sup jBðyÞ  Bð1ÞjXh pffiffiffiffiffiffiffiffi ¼ P sup jBðuÞjX pffiffiffiffiffiffiffiffi Mm Mm jy1jpd u2½0;2d  hW m ¼ P sup jBðuÞjX pffiffiffiffiffiffiffiffiffiffiffiffiffi 2dM m 0pup1    hW m ¼ P max sup BðuÞ; sup ðBðuÞÞ X pffiffiffiffiffiffiffiffiffiffiffiffiffi 2dM m 0pup1 0pup1    hW m hW m p2P sup BðuÞX pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4C pffiffiffiffiffiffiffiffiffiffiffiffiffi . 2dM m 2dM m 0pup1 Consequently,         Wm hW m Wm hW m C ðe þ hÞ pffiffiffiffiffiffiffiffi  4C pffiffiffiffiffiffiffiffiffiffiffiffiffi  Dm pPfjS m j4eW m gpC ðe  hÞ pffiffiffiffiffiffiffiffi þ 4C pffiffiffiffiffiffiffiffiffiffiffiffiffi þ Dm . Mm Mm 2dM m 2dM m This completes the proof.

&

We shall apply Lemma 8 to triangular arrays. Let again X ¼ fX n;k ; 1pkpkn ; nX1g be a triangular array of real centered independent random variables and a ¼ fan;k ; 1pkpkn ; nX1g a triangular array of positive reals. By considering, if necessary, a larger probability space, we can always assume that there exists a sequence n1 ; n2 ; . . . such that for each positive integer n, nn ¼ fxn;k ; kX1g;

with xn;k ¼ X n;k ; 1pkpkn ,

n

and n is a sequence of independent random variables. Further the sequences n1 ; n2 ; . . . are mutually independent. By suitably enlarging the probability space, there exists for each integer n a linear Brownian motion Bn ¼ fBn ðtÞ; 0pto1g starting at 0 and a sequence tn1 ; tn2 ; . . . of independent non-negative random variables with Etnk ¼ a2n;k Ex2n;k , kX1 such that, with tn0 ¼ 0 a.s., ( ! ! ) k k1 X X D tnj  Bn tnj ; kX1 ¼fan;k xn;k ; kX1g. Bn j¼0

j¼0

In fact, in each step, it would be enough to let k run between 1 and kn . By applying Lemma 8 with the choice n ¼ nn , m ¼ kn , we now easily deduce the following corollary.

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pffiffiffiffiffi Corollary 9. Let e; h; d be positive reals with e4h4 2d. Then, with notation (1), for n ¼ 1; 2; . . .     h h Cððe þ hÞC n Þ  4C pffiffiffiffiffi C n  Dn ðdÞpPfjT n j4eAn gpCððe  hÞC n Þ þ 4C pffiffiffiffiffi C n þ Dn ðdÞ, 2d 2d where

 ( ) kn  X   tn  B2n XdB2n . Dn ðdÞ ¼ P    j¼0 j

This result will allow us to establish the following statement. Proposition 10. Assume that X and a satisfy X Dm ðdÞo1 for all d40.

(2)

m

Then c:c:

T n =An ! 0 () LðaÞ ¼ 0. a:s:

This proposition can be viewed as an extension of Theorem 1, since in the Gaussian case tnj ¼ a2n;j . ffiC Þ, which is achieved by using Proof. The key lies in the comparison between Cððe þ hÞC n Þ and Cðphffiffiffi 2d n c:c: Lemma 7. The implication LðaÞ ¼ 0 ) T =A ! 0 is easy. Indeed, if LðaÞ ¼ 0, then for any r40 the series n n P rC 2 n converges, or equivalently, e n X CðrC n Þo1 for all r40. (3) n

pffiffiffiffiffi Let e40, and choose h; d in Corollary 9 such that h ¼ e=24 2d. By Corollary 9 and the assumption made, the P series n PfjT n j4eAn g converges provided  X X  h Cððe  hÞC n Þo1; C pffiffiffiffiffi C n o1. 2d n n P And this holds true if n Cððe=2ÞC n Þo1, which is satisfied by assumption. Hence the first part of Proposition 10 is proved. P c:c: Conversely, if T n =An ! 0, then the series n PfjT n j4eAn g converges for any e40. We shall prove that (3) holds true. We distinguish two cases. Case choose e; h; d such as e ¼ h ¼ r=2, d ¼ 1=8, so that pffiffiffiffiffi 1: lim inf n!1 C n ¼ 1. Let r40 be fixed, wepffiffiffiffiffi h= 2d ¼ 2r. Then Cððe þ hÞC n Þ ¼ CðrC n Þ and Cððh= 2dÞC n Þ ¼ Cð2rC n Þ. By Lemma 7, CðrC n Þ 

2 1 eðrC n Þ =2 ; 1 þ rC n

Cð2rC n Þ 

2 1 e2ðrC n Þ , 1 þ 2rC n

so that, for any ror1 or2 o2r, if n is sufficiently large CðrC n ÞXeðr1 C n Þ

2

=2

;

4Cð2rC n Þpeðr2 C n Þ

2

=2

.

Therefore, CðrC n Þ  4Cð2rC n ÞXeðr1 C n Þ

2

=2

2

2

ð1  eðr2 r1 ÞðC n Þ

2

=2

ÞXð1=2Þeðr1 C n Þ

2

=2

P 2 for n sufficiently large. In view of Corollary 9, and assumption (2) this implies that the series n eðr1 C n Þ =2 converges. This being true for any r40 and any r1 4r, it follows that (3) is satisfied, as claimed. Case 2: lim inf n!1 C n o1. In this case there exist a sequence of indices fnj ; jX1g and a real t such that limj!1 C nj ¼ t. Choose r40 such that CðrtÞ44Cð2rtÞ, and let again e; h; d such as e ¼ h ¼ r=2, d ¼ 1=8. Applying Corollary 9 for n ¼ nj , j ¼ 1; 2; . . . gives CðrC nj Þ  4Cð2rC nj ÞpPfjT nj j4eAnj g þ Dnj ðdÞ.

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Letting now j tend to infinity implies 0oCðrtÞ  4Cð2rtÞp lim inf ðPfjT nj j4eAnj g þ Dnj ðdÞÞ, j!1

which contradicts the fact that both series

P

n PfjT n j4eAn g,

P

n Dn ðdÞ

converge. The proof is now complete.

&

We can now pass to the proof of Theorem 6. Let pX2 and let a ¼ fan;k ; 1pkpkn ; nX1g be a triangular array of positive reals such that b ¼ a2 is p-regular. Let X ¼ fX n;k ; 1pkpkn ; nX1g be a triangular array of real centered iid random variables with finite 2p-th moment. We shall make use of the fact (Fisher (1992), D Theorem 2.1) that for each n, we can assume that ftnk ; 1pkpkn g ¼fa2n;k ynk ; 1pkpkn g, and fynk ; 1pkpkn g is an iid sequence with finite pth moments. As b is p-regular, (2) is satisfied. Using Proposition 10, we get the desired conclusion. & Remark. Although the characterization given in Theorem 6 is simple, it is rather abstract. Usually condition (2) is as difficult to check as the fact that a is 2p-regular. Thus the interest in a statement like Theorem 6 is the link established between p-regularity and 2p-regularity, via the arrays a and b. It is possible to check directly condition (2), by imposing conditions on the weights, which, however, appear to be stronger than the condition LðaÞ ¼ 0. To see this, we shall use some arguments from Weber (2006). In order to avoid unnecessarily heavy notation, we simply return to the setting considered in Lemma 8, and will bound the quantity  ( )  X m   Dm ¼ Dm ðdÞ ¼ P  t  M m XdM m .   j¼0 j Using inequality (1.2) in Davis (1976) we see that if Ejxi j2þe o1 for some e40, the sequence of stopping times ti satisfies 1þe=2

Eti

2þe pCw2þe , i Ejxi j

(4)

where the constant C depends on e only. Let pX2. Assume that for any positive integer j, xi 2 L2p , and moreover Qp ðnÞ:¼ sup kxi kp o1. jX1

Put for any positive integer l, xl ¼ tl  Etl ¼ tl  w2l . Then using (4) with 2ðp  1Þ ¼ e gives 2p 0 p Ejxl jp p2p ðEjtl jp þ w2p l ÞpC p ð1 þ Qp ðnÞÞwl ,

where C 0p depends on p only, and may vary in the next lines. Further note that in the case xl 2 L4 , lX1 we have 0pEx2l ¼ Et2l  ðEtl Þ2 pEt2l pC 02 w4l Ejxl j4 . Apply now Rosenthal’s inequality (see e.g. Petrov, 1995, p. 59). In view of centering and independence of the xl ’s, we get 0 p  !p=2 1  X m m m X X   A E ðtl  w2l Þ pC 0p @ w2p Ex2l l þ   l¼1 l¼1 l¼1 0 !p=2 1 !p=2 m m m X X X 2p 0 p 4 0 p 4 @ A pC p ð1 þ Qp ðnÞÞ pC p ð1 þ Qp ðnÞÞ wl þ wl wl . l¼1

l¼1

l¼1

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Consequently, by using Chebyshev’s inequality,  ( ) Pm 4 p=2 X  m   0 p l¼1 wl t  M m XdM m pC p ð1 þ Qp ðnÞÞ . Dm ðdÞ ¼ P   j¼0 j  ðdM m Þ2 We thus see that condition (2) holds provided !p X ½Pm w4 1=2 l¼1 l o1. m Mm For triangular arrays, this means that P n 4 1=2 !p X ½ kk¼1 an;k  o1, Pkn 2 n k¼1 an;k establishing Theorem 2. As we noted earlier, LðaÞ ¼ 0 is equivalent to 0  2 1 Pk n X k¼1 an;k B C exp@d Pkn Ao1 for all d40. 2 a n k¼1 n;k Proof of Theorem 3. Since X is symmetric, it has the same law as X ¼ fen;k X n;k ; 1pkpkn ; nX1g, where e ¼ fen;k ; 1pkpkn ; nX1g is a Rademacher sequence defined on a joint probability space ðOe ; Ae ; Pe Þ (with corresponding expectation symbol Ee ). Put Pk n 2 2 kn X k¼1 an;k X n;k Yn ¼ an;k en;k X n;k ; Qn ¼ . B2n k¼1 Let fOn ; nX1g be a sequence of positive reals. Write    jT n j jY n j jY n j P 4e ¼ EPe 4e pPfQn 4On g þ E1fQn pOn g Pe 4e . An An An Further, there exists an absolute constant C such that ( ) ( )   jY n j e2 A2n e2 A2n e2 C 2n Pe 4e p exp C Pkn . ¼ exp C ¼ exp C 2 2 An Qn Qn B2n k¼1 an;k X n;k We deduce that   jT n j e2 C 2n 4e pPfQn 4On g þ exp C P . An On It follows that if 1 X

ðaÞ then

PfQn 4On go1;

n¼1 c:c: T n =An ! 0.

On ¼

C 2n =ðL

ðbÞ

log nÞ, c:c:

n¼1



Choosing in particular (with L41)

shows that T n =An ! 0, provided that 1 X

e2 C 2n exp C o1 On n¼1

1 X

PfQn 4lC 2n = log ngo1

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for any l40. To connect the last sum with the sum in Theorem 3, we use Rosenthal’s inequality. Recall that we assumed for 1pkpkn ; nX1 that EX 2n;k ¼ 1, and for some pX2, X n;k 2 L2p . Put Y n;k ¼ a2n;k ðX 2n;k  1Þ;

1pkpkn ; nX1,

then for sufficiently large n we have (P k PfQn 4lC 2n = log

2 n 2 k¼1 an;k X n;k 2 Bn

ng ¼ P ( pP

kn X

a2n;k ðX 2n;k

k¼1

A2 4l 2 n Bn log n

(

) ¼P

kn X

) a2n;k X 2n;k 4lA2n

log n

k¼1

) ! Pn Ej kk¼1 Y n;k jp l 2  1Þ4 An log n p l 2 . 2 ð2 An log nÞp

Now, by Rosenthal’s inequality 9 8 p   !2 p=2  > p > kn k k  X = < n n X   X p   p   , Y n;k  p C 0 E Y þ EjY j E n;k n;k     k¼1 > log p >  ; : k¼1 k¼1 where C 0 is an absolute constant. But 2  kn kn kn  X X X   E Y n;k  ¼ a4n;k EðX 2n;k  1Þ2 pCkX k44 a4n;k ,   k¼1 k¼1 k¼1 so that 8 9 p   !p=2 p < kn kn kn  X = X X p   2p E Y n;k  p C 0 CkX k44 a4n;k þ kX k2p a 2p n;k ;   k¼1 log p : k¼1 k¼1 pC p maxðkX k44 ; kX k2p 2p Þ

kn X

!p=2 a4n;k

.

k¼1

Therefore, PfQn 4lC 2n = log

ngpC p maxðkX k44 ; kX k2p 2p Þ

Pkn

4 p=2 k¼1 an;k Þ ðl2A2n log nÞp

ð

! .

This completes the proof of Theorem 3. & Proof of Theorem 4. Let Y 1 ; . . . ; Y n be independent symmetric random variables, S n ¼ Y 1 þ    þ Y n . One part of the Hoffmann-Jørgensen (1974) inequality states that  p p PfjSn j43 tgpC p P max jX k j4t þ C p fPðjSn j4tÞg2 (5) 1pkpn

for any integer pX1, where C p is a constant depending on p. By (5) we have PfjT n j43p eAn gpDC p

kn X

p

PfjDak X j4eAn g þ C p ðPfjT n j4eAn gÞ2 .

k¼1

Choosing p large enough and summing (6) for n ¼ 1; 2; . . . we get 1 X n¼1

PfjT n j43p eAn gpDC p

X 1pkpkn nX1

PfjX j4eAn =Dak g þ C p

1 X p ðPfjT n j4eAn gÞ2 . n¼1

(6)

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Assumptions (a) and (b) therefore imply (c). Conversely if (c) is true, then 

  1 1 PfjT n j4eAn gX P max jak X k jXeAn ¼ 1  P max jak X k joeAn 1pkpkn 1pkpkn 2 2 " # " # kn kn Y Y 1 1 1 ¼ ð1  Pfjak X k jXeAn gÞ X 1  ePfjak X k jXeAn g 2 2 k¼1 k¼1

 Pk n 1 1  Pfja X jXeA g l n k k k¼1 ¼ :¼ ½1  e n . 1e 2 2 From this estimate and (c) follows that ln tends to 0, and then the chain of estimates can be continued as 1 2½1

 eln  ¼ 12½ln þ Oðl2n ÞX14ln ,

for any integer n sufficiently large. Therefore, for n large PfjT n j4eAn gX14ln . And consequently (c) implies

P

n

ln o1, which is exactly (a).

&

Proof of Theorem 5. The proof is based on a convexity argument enabling us to use the Gaussian randomization technique. First of all, there is no loss of generality in assuming that for any nX1 and 1pkpkn we have jX n;k jp1 a.s. Let X0 be an independent copy of XPdefined on a joint probability space ðO0 ; A0 ; P0 Þ with corresponding n expectation symbol E0 . Write T 0n ¼ kk¼1 an;k X 0n;k . Let e ¼ fen;k ; 1pkpkn ; nX1g be a triangular array of independent Rademacher random variables defined on a joint probability space ðOe ; Ae ; Pe Þ, with corresponding expectation symbol Ee . Similarly, let g ¼ fgn;k ; 1pkpkn ; nX1g be a triangular array of independent Nð0; 1Þ distributed random variables defined on a joint probability space ðOg ; Ag ; Pg Þ, with corresponding expectation symbol Eg . Let A be any real number and consider the convex non-decreasing function jA ðxÞ ¼ ðx  AÞþ . If X is any random variable, then for any positive real a, aPfX 4A þ agpEjA ðX Þ. Applying this for A ¼ An e ¼ a and X ¼ T n and then using Jensen’s inequality lead to ðeAn ÞPfT n 42eAn gpEjeAn ðT n Þ ¼ EjeAn ðT n  E0 T 0n Þ pEE jeAn ðT n  0

T 0n Þ

¼ EEe jeAn

! an;k en;k X n;k

k¼1

Pk n

k¼1 an;k en;k ðEg jgn;k jÞX n;k 1=2

¼ EEe jeAn

ð2=pÞ

pEEe Eg jeAn

Pk n

k¼1 an;k en;k jgn;k jX n;k 1=2

ð2=pÞ

Pk n ¼ EEg jeAn

kn X

k¼1 an;k gn;k X n;k 1=2

ð2=pÞ

!

!

! .

ð7Þ D

In the last equality we used the fact that fen;k jgn;k j; 1pkpkn ; nX1g ¼fgn;k ; 1pkpkn ; nX1g. Applying it now to A ¼ An e ¼ a and X ¼ T n , and arguing similarly also gives ! Pk n k¼1 an;k gn;k X n;k ðeAn ÞPfT n 42eAn gpEEg jeAn . (8) ð2=pÞ1=2

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962

As PfjT n j42eAn gpPfT n 42eAn g þ PfT n 42eAn g, we obtain from (7) and (8) ! Pk n k¼1 an;k gn;k X n;k ðeAn ÞPfjT n j42eAn gp2EEg jeAn . ð2=pÞ1=2 But, Pk n Eg jeAn

k¼1 an;k gn;k X n;k 1=2

!

) ð2=pÞ1=2 u P Nð0; 1Þ4 du Vn eAn Z 1 Vn pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi PfNð0; 1Þ4vg dv 2=p ð 2=peAn Þ=V n Z 1 Z 1 Vn dw 2 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ew =2 pffiffiffiffiffiffi dv 2p 2=p ð 2=peAn Þ=V n v ! pffiffiffiffiffiffiffiffi rffiffiffi Z 1 2=peAn e2 A2n =pV 2n Vn p Vn v2 =2 R RðvÞe dvp e ffi 2 2 2 ðpffiffiffiffiffi Vn 2=peAn Þ=T n

Z ¼

ð2=pÞ

¼ ¼ ¼ p

1

(

pV n e2 A2n =pV 2n e . 4

Therefore, PfjT n j42eAn gpE

pV n e2 A2n =pV 2n e . 4eAn

We now make use of the boundedness assumption on the sequence X. The above inequality becomes in this case 2 p 2 2 PfjT n j42eAn gp Eee An =pV n , 4e Pn 2 2 Pn 2 since V 2n ¼ kk¼1 an;k X n;k p kk¼1 an;k pA2n a.s. Put for m ¼ 1; 2; . . . : J m ¼ fn : mpAn =V n om þ 1g. Then 1 X

1 1 X 2 pX pX 2 2 2 2 2 2 E ee An =pV n p E½]fJ m gee m =2p ee m =2p  ðe2 m2 =pÞ 4e m¼1 n2J 4e m¼1 m " # 1 X p 2 2 2 2 p E sup ½]fJ m gee m =2p  ee m =2p 4e mX1 m¼1

PfjT n j42eAn gp

n¼1

pC e E sup mX1

]fn : moAn =V n pm þ 1g . expfe2 m2 =2pg

This completes the proof of Theorem 5. & Acknowledgment We thank Anna Kuczmaszewska for several useful remarks concerning the first draft of the paper. References Adler, A., Cabrera, M., Rosalsky, A., Volodin, A., 1999. Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces. Bull. Inst. Math. Acad. Sinica 27, 187–212. Ahmed, S., Antonini, R.G., Volodin, A., 2002. On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes. Statist. Probab. Lett. 58, 185–194. Baum, L.E., Katz, M., 1965. Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108–123.

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963

Breiman, L., 1968. Probability. Addison-Wesley, Reading, MA. + M., Re´ve´sz, P., 1981. Strong Approximations in Probability and Statistics. Academic Press, New York. Cso¨rgo, Davis, B., 1976. On the Lp norms of stochastic integrals and other martingales. Duke Math. J. 43, 697–704. + P., 1949. On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291. Erdos, Fazekas, I., 1985. Convergence rates in the Marczinkiewicz strong law of large numbers for Banach space values random variables with multidimensional indices. Publ. Math. Debrecen 32, 203–209. Fazekas, I., 1992. Convergence rates in the law of large numbers for arrays. Publ. Math. Debrecen 41, 53–71. Fisher, E., 1992. A Skorohod representation and an invariance principle for sums of weighted iid random variables. Rocky Mount. J. Math. 22, 169–179. Gut, A., 1992. Complete convergence for arrays. Period. Math. Hungar. 25, 51–75. Hoffmann-Jørgensen, J., 1974. Sums of independent Banach space valued random variables. Studia Math. 52, 159–186. Hu, T.-C., Mo´ricz, F., Taylor, R.L., 1989. Strong laws of large numbers for arrays of rowwise independent random variables. Acta Math. Hungar. 54, 153–162. Hu, T.-C., Rosalsky, A., Szynal, D., Volodin, A., 1999. On complete convergence for arrays of rowwise independent random elements in Banach spaces. Stochastic Anal. Appl. 17, 963–992. Hsu, P.L., Robbins, H., 1947. Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25–31. Kuczmaszewska, A., Szynal, D., 1988. On the Hsu–Robbins law of large numbers for subsequences. Bull. Polish Acad. Sci. Math. 36, 69–79. Kuczmaszewska, A., Szynal, D., 1990. On complete convergence for partial sums of independent identically distributed random variables. Probab. Math. Statist. 11, 223–235. Kuczmaszewska, A., Szynal, D., 1994. On complete convergence in a Banach space. Internat. J. Math. Math. Sci. 17, 1–14. Li, D., Rao, M.B., Wang, X., 1992. Complete convergence of moving average processes. Statist. Probab. Lett. 14, 111–114. Mitrinovic´, D.S., 1970. Analytic inequalities. Die Grundlehren der Mathematischen Wissenschaften, Band 1965, Springer, Berlin. Petrov, V.V., 1995. Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Clarendon Press, Oxford. Pruitt, W., 1966. Summability of independent random variables. J. Math. Mech. 15, 769–776. Rao, M.B., Wang, X., Yang, X., 1993. Convergence rates on strong laws of large numbers for arrays of rowwise independent elements. Stochastic Anal. Appl. 11, 115–132. Rohatgi, V.K., 1971. Convergence of weighted sums of independent random variables. Proc. Cambridge Philos. Soc. 69, 305–307. Sung, S.H., 1997. Complete convergence for weighted sums of arrays of rowwise independent B-valued random variables. Stochastic Anal. Appl. 15, 255–267. Weber, M., 1995. Borel matrix. Comment. Math. Univ. Carolin. 36, 401–415. Weber, M., 2006. A weighted CLT. Statastic. Probab. Lett. 76, 1482–1487.