Marcinkiewicz-Zygmund Type Law of Large Numbers for Double Arrays of Random Elements in Banach Spaces

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1 ISSN , Lobachevskii Journal of Mathematics, 2009, Vol. 30, No. 4, c Pleiades Publishing, Ltd., Marcinkiewicz-Zygmund Tye Law of Large Numbers for Double Arrays of Random Elements in Banach Saces Le Van Dung *, Thuntida Ngamkham 2, Nguyen Duy Tien **,and A.I.Volodin 3*** Faculty of Mathematics, National University of Hanoi, 3 34 Nguyen Trai, Hanoi, Vietnam 2 Deartment of Mathematics and Statistics, Thammasat University, Rangsit enter, Pathumthani 22, Thailand 3 School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, rawley, WA 6009, Australia Received July 30, 2009 Abstract In this aer we establish Marcinkiewicz-Zygmund tye laws of large numbers for double arrays of random elements in Banach saces. Our results extend those of Hong and Volodin [6] Mathematics Subject lassification: 60B, 60B2, 60F5, 60G42 DOI: 0.34/S Key words and hrases: Marcinkiewicz-Zygmund inequality, Rademacher tye Banach saces, Martingale tye Banach saces, Double arrays of random elements, Strong and L laws of large numbers.. INTRODUTION Marcinkiewicz-Zygmund tye strong laws of large numbers were studied by many authors. In 98, Etemadi [3] roved that if {X n ; n } is a sequence of airwise i.i.d. random variables with EX <, then lim n i= n X i EX )=0a.s. Later, in 985, hoi and Sung [2] have shown that if {X n ; n } are airwise indeendent and are dominated in distribution by a random variable X with E X log + X ) 2 <, <<2, then lim n X n i EX i )=0a.s. i= Recently, Hong and Hwang [5], Hong and Volodin [6] studied Marcinkiewicz-Zygmund strong law of large numbers for double sequence of random variables, Quang and Thanh [2] established the Marcinkiewicz-Zygmund strong law of large numbers for blockwise adated sequence. In this aer, we extend the results of Hong and Volodin [6] to some secial class of Banach saces, so-called Banach saces that satisfy the maximal Marcinkiewicz-Zygmund inequality with exonent see the definition below). This class includes Rademacher tye and martingale tye Banach saces, 0 < 2. For a, b R, max {a, b} will be denoted by a b. Throughout this aer, the symbol will denote a generic constant 0 << ) which is not necessarily the same one in each aearance. * lvdunght@gmail.com ** ndtien@it-hu.ac.vn *** address: andrei@maths.uwa.edu.au 337

2 338 DUNG et al. 2. PRELIMINARIES Technical definitions relevant to the current work will be discussed in this section. The Banach sace X is said to be of Rademacher tye 2) if there exists a constant < such that n n E V j E V j for all indeendent X -valued random elements V,...,V n with mean 0. We refer the reader to Pisier [0] and Woyczyński [6] for a detailed discussion of this notion. Scalora [4] introduced the idea of the conditional exectation of a random element in a Banach sace. For a random element V and sub σ-algebra G of F, the conditional exectation EV G) is defined analogously to that in the random variable case and enjoys similar roerties. A real searable Banach sace X is said to be martingale tye 2) if there exists a finite ositive constant such that for all martingales {S n ; n } with values in X, su E S n n E S n S n. n= It can be shown using classical methods from martingale theory that if X is of martingale tye,then for all r< there exists a finite constant such that ) r E su S n r E S n S n. n n= learly every real searable Banach sace is of martingale tye and the real line the same as any Hilbert sace) is of martingale tye 2. If a real searable Banach sace of martingale tye for some < 2then it is of martingale tye r for all r [,). It follows from the Hoffmann-Jørgensen and Pisier [4] characterization of Rademacher tye Banach saces that if a Banach sace is of martingale tye, then it is of Rademacher tye. But the notion of martingale tye is only suerficially similar to that of Rademacher tye and has a geometric characterization in terms of smoothness. For roofs and more details, the reader may refer to Pisier [0, ]. Definition. Let 0 < 2. We say that a collection {V ij ; i m, j n} of random elements taking values in a real searable Banach sace X satisfies the maximal Marcinkiewicz- Zygmund inequality with exonent if k l m n E max k m V ij, 2.) l n where the constant is indeendent of m and n. It is clear that for 0 <, every collection random elements {V ij ; i m, j n} satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent. Now we rovide an examle of a collection of random elements {V ij ; i m, j n} taking values in a real searable Banach sace X that satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent in the case of < 2. Examle. Suose that {V j ; j n} is a collection of n indeendent mean 0 random elements taking values in a Rademacher tye < 2) Banach sace. Then there exists a constant such that n n E V j E V j.

3 MARINKIEWIZ-ZYGMUND TYPE LAW 339 This follows from the following observation. If we set V j = V j and V ij =0 for all 2 i m, j n,then{v ij ; i m, j n} satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent by the definition of the Rademacher tye. Random elements {V ; m,n } are said to be stochastically dominated by a random element V if for some constant < P { V >t} P{ V >t}, t 0, m, n. Denote d k be the number of divisors of k. The following lemma can be found in Gut and Sǎtaru [7] or Gut [8]. Lemma. n d k k γ = On γ log n) γ<), 2.2) k= k=i+ ) d k log i k γ = O i +) γ γ >). 2.3) 3. MAIN RESULTS With the reliminaries accounted for, the main results may now be established. In the following we let {V ; m,n } be an array of random elements in a real searable Banach sace X. Theorem. Let 0 < 2 and {V ij ; i,j } be an array of random elements that satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent.if i α j β ) < for some α>0and β>0, then m m α n β and and Proof. Set m α n β m T kl = n V ij 0 a.s. as m n 3.) n V ij 0 in L as m n. 3.2) S = max 2 k m<2 k+ 2 l m<2 l+ m n V ij, S m α n β S 2 k 2 l 2 kα 2 lβ. First, we rove 3.). For arbitrary ε>0, by the Markov and maximal Marcinkiewicz-Zygmund inequalities with exonent we have { } S P 2 k 2 l 2 kα 2 lβ >ε 2 kα 2 lβ ) ε E S 2 k 2 l k= l= k= l= 2 kα 2 lβ ) ε 2 k 2 l k= l= ε i α j β ) <.

4 340 DUNG et al. It follows by the Borel-antelli lemma that lim k l S 2 k 2 l 2 kα =0 a.s. 3.3) 2lβ Again, let ε>0be arbitrary, from the maximal Marcinkiewicz-Zygmund inequality with exonent we obtain { S2 P { T kl >ε} P k 2 l 2 kα 2 lβ > ε } + P 2 max S 2 k m<2 k+ m α n β > ε { 2 P S2 k 2 l 2 kα 2 lβ > ε } 2 2 l n<2 l+ + P max S 2 k m<2 k+ 2 kα 2 lβ > ε { 2 P S 2 k 2 l > 2kα 2 lβ } ε + P 2 max 2 kα 2 lβ ε S > m 2 k+ 2 2 l n<2 l+ n 2 l+ 2 2 kα 2 lβ ) ε E S 2 k 2 l kα 2 lβ ) ε 2 k+ i= 2 l+ 2 2 kα 2 lβ ) ε E max S m 2 k+ n 2 l+ 2k ε From this we obtain P { T kl >ε} k= l= k= l= 2k+ ε ε i= 2 k+ i= 2 l 2 l+ 2 l+ 2 kα 2 lβ ) + ε 2 k+)α 2 l+)β ). 2 2 kα 2 lβ ) ε 2 k+ i= 2 k+)α 2 l+)β ) ε 2 l+ 2 k 2 l 2 k+)α 2 l+)β ) i α j β ) <. Again by the Borel-antelli lemma, we have that lim kl =0 k l a.s. 3.4) Note that for 2 k m<2 k+ and 2 l n<2 l+, S m α n β S m n β S 2 k 2 l 2 kα 2 lβ 2 k 2 l 2 kα 2 lβ T kl + S 2 k 2 l 2 kα, 3.5) 2lβ and so the conclusion 3.) follows from 3.3) and 3.4). Next, we will rove 3.2). Since convergence of the following series we get k= l= 2 kα 2 lβ ) E S 2 k 2 l On the other hand, we have that E T kl E + max 2 k m<2 k+ 2 l m<2 l+ 2 kα 2 lβ ) E S 2 k 2 l k= l= 2 kα 2 lβ ) 2 k 2 l i α j β ) <, lim E S 2 k 2 l k l 2 kα 2 lβ =0. 3.6) S m α n β + E S 2 k 2 l 2 kα 2 lβ ) 2 k+ i= 2 kα 2 lβ ) 2 kα 2 lβ ) E max m 2 k+ n 2 l+ + 2 l+ 2 kα 2 lβ ) 2 k 2 l S

5 2 k 2 l This imlies E T kl k= l= MARINKIEWIZ-ZYGMUND TYPE LAW 34 2k+ 2 kα 2 lβ ) + k= l= i= 2 k+ i= 2 l+ 2k+ 2 k+)α 2 l+)β ) 2 l+ i= 2 k+)α 2 l+)β ) 2 l+ 2 k+)α 2 l+)β ). i α j β ) <. Hencewehave lim E T kl =0. 3.7) k l It follows from 3.5) that for 2 k m<2 k+, 2 l n<2 l+, E S m α n β E T kl + E S 2 k 2 l 2 kα 2 lβ ). Thus, the conclusion 3.2) follows from 3.6) and 3.7). In the next two theorems, we obtain the Marcinkiewicz-Zygmund tye law of large numbers for double array of random elements. Theorem 2. Let <r< 2 and {V ; m,n } be an array of random elements that satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent. Suose that {V ; m,n } is be stochastically dominated by a random element V such that E V r log + V r < if r<then m n V ) ij 0 a.s. and in L as m n. 3.8) r Proof. Let F be the distribution of V. Set V ij = V iji V ij ij) r ),V ij = V iji V ij > ij) r ). Alying the equation 2.3) of Lemma 2. with γ = r we obtain d k log i k=i+ = O ). Then k r i +) r we have E V ij ij) r d k k r k r k= 0 i=0 x df x) log i i +) r i+) r i r i=0 d k k r k=i+ x df x) E V r log + V r <. ) i+) r i r x df x) 3.9) On the other hand, alying the equation 2.2) of Lemma 2. with γ = r we obtain n d k k= On r log n). Hence we have the following inequalities E V ij ij) r d k k= k r k r x df x) k k= d i i= i r ) i+) r i r x df x) k r =

6 342 DUNG et al. i r log i i= This imlies that 3.9) and 3.0) yield i+) r i r x df x) x r log + x r df x) E V r log + V r <. E V ij ij) r E V ij <. ij) r <. 3.0) Alying Theorem 3. with α = β = we obtain 3.8). r Theorem 3. Let 0 <r<. If{V ; m,n } is be stochastically dominated by a random element V such that E V r log + V r <, then m n V ) ij 0 a.s. and in L as m n. 3.) r Proof. By 2.3) of Lemma 2. with γ = r we obtain d k k=i+ show that k r E V ij E V r log + V r <. ij) r ) log i = O. Hence we can i +) r Alying Theorem 3. with α = β = we obtain 3.). r In the next sections we resent corollaries to the theorems. 4. MARTINGALE TYPE BANAH SPAE ASE The fact the a collection of a martingale difference random elements {V ij ; i m, j n} taking values in martingale tye satisfy the maximal Marcinkiewicz-Zygmund inequality with exonent, follows from the lemma below. We denote F kl the σ-field generated by the family of random elements {V ij ; i<kor j<l}, F, = { ;Ω}. Lemma 2. Let 2 and let {V ij ; i m, j n} be a collection of random elements in a real searable martingale tye Banach sace with EV ij F ij )=0for all i m, j n.then{v ij ; i m, j n} satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent. Proof. The conclusion 2.) is trivial in the case of =. Hence we consider the case of < 2. SetS kl = k l V ij,y l =max k m S kl. Ifσ l is a σ-field generated by {V ij ; i m, j l} then for each l l n), σ l F i,l+ for all i. This imlies that EV i,l+ σ l )= EEV i,l+ F i,l+ ) σ l )=0. Thus, we have k ES k,l+ σ l )=ES kl σ l )+ EV i,l+ σ l )=S kl, i=

7 MARINKIEWIZ-ZYGMUND TYPE LAW 343 that is, {S kl,σ l ; l n} is a martingale. Hence, { S kl,σ l ; l n} is a nonnegative submartingale for each k =, 2,...,m. By Lemma 2.2 of Thanh [3] it follows that {Y l,σ l ; l n} is a nonnegative submartingale. Alying Doob s inequality see, e.g., how and Teicher [,. 255]), we obtain E ) max S kl = E k m l n max l n Y l ) EY n. 2.2) On the other hand, since EV ij F ij )=0we have that {S kn, G k = F k+, ; k m} is a martingale. Thus m n EYn = E max S kn E V kj. 2.3) k m Note again that for each k k m), { l V kj, G kl = F k,l+ ; l n} is a martingale. Hence, E n V kj E max l n k= l V kj n E V kl. 2.4) ombining 2.2), 2.3) and 2.4) we obtain 2.). The following theorem characterizes the martingale tye Banach saces. Theorem 4. Let {V ; m,n } be an array of random elements in a real searable Banach sace X. Then the following two statements are equivalent: i) The Banach X is of martingale tye. ii) For every double arrays {V ; m,n } of random elements in X with EV F )= 0 for all m,n, the condition E V m α n β ) < m= n= for some α>0and β>0, imlies that m n m α n β V ij 0 a.s. as m n. Proof. First we rove [i) ii)]. The case of <r< 2 follows from Lemma 5. and Theorem 3.. Hence we only rove for the case of r =. Alying 2.3) from Lemma 2. with γ =, we obtain ) d k log i k=i+ k = O i +). Hencewehave i=0 k=i+ E V ij EV ij F ij) ij) ) i+) d k k x df x) i i=0 log i i +) E V ij ij) i+) i l= k= d k k k x df x) 0 x df x) E V log + V <. By Theorem 3. with α = β = m n V ij EV ij F ij )) 0 a.s. as m n.

8 344 DUNG et al. Next, for arbitrary ε>0, P V ij >ε) d k P V >k)= i log i i= Hence by the Borel-antelli lemma Finally, by the identity m n V ij = it it enough to show that m m k= i+ i i ) i+ d k df x) i= k= df x) E V log + V <. m n V ij 0 n V ij EV ij F ij )+ a.s. m n The roof of 4.4) is the same as that of m i= of Hong and Volodin for = ) with noting that V ij + i m n EV ij F ij ), n EV ij F ij ) 0 a.s. as m n. 4.4) n [EX ij F ij) EX ij F ij)] 0 a.s. as m n EV ij F ij) E V ij F ij) 2E V I V >ij) F ij ), and we use V ij and V are instead of X ij and X, resectively. Now we rove [ii) i)]. Assume that ii) holds. Let {W n, G n ; n } be an arbitrary sequence of martingale difference in X such that E W n < n= n For n, set V = W n if m = and V =0 if m 2. Then {V ; m,n } is a double array of random elements in in X satisfies EV F )=0for all m,n, and E V E W n ) = n <. By ii), m= n= m n= n V ij 0 a.s. as m n. Taking m =and letting n we obtain n W j 0 n a.s. as n. Thenby Theorem 2.2 of Hoffmann-Jørgensen and Pisier [4], X is of martingale tye. The next corollary follows from Theorems 3.2 and 3.3.

9 MARINKIEWIZ-ZYGMUND TYPE LAW 345 orollary. Let r< 2 and X be a martingale tye Banach sace. Suose that {V ; m,n } is be stochastically dominated by a random element V such that E V r log + V r < if r< m n V ) ij EV ij F ij )) 0 r a.s. and in L r as m n. 5. RADEMAHER TYPE BANAH SPAE ASE The fact that a collection {V ij ; i m, j n} of indeendent mean zero random elements saces taking values in Rademacher tye satisfy the maximal Marcinkiewicz-Zygmund inequality with exonent, follows from the following lemma. For the roof we refer to Lemma 2.3. of Thanh [3], while it is obvious. Lemma 3. Let {V ij ; i m, j n} be a collection of indeendent mean 0 random elements in a real searable Rademacher tye Banach saces. Then {V ij ; i m, j n} satisfies the maximal Marcinkiewicz-Zygmund inequality with exonent. The following theorem can be roved in the same way as Theorem 4.2 and hence we omit its roof. Theorem 5. Let {V ; m,n } be an array of indeendent random elements in a real searable Banach sace X. Then the following two statements are equivalent: i) The Banach X is of Rademacher tye. ii) For every double arrays {V ; m,n } of indeendent mean 0 random elements in X, the condition i α j β ) < for some α>0andβ >0 imlies that m n m α n β V ij 0 a.s. as m n. Note that the above result is more general than Theorem 3. necessity art) of Rosalsky and Thanh []. An oen roblem. A relatively interesting case r =is not considered in Theorem 3.2, while we roose that the result remains true in this case. REFERENES. Y. S. hao and H. Teicher, Probability Theory. Indeendence, Interchangeability, Martingale Sringer, New York, 997). 2. B. D. hoi, and S. H. Sung, On convergence of S n ES n )/n /r, <r<2 for airwise indeendent random variables, Bull. Korean Math. Soc. 22 2), ). 3. N. Etemadi, An elementary roof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 ), 9 98). 4. J. Hoffmann-Jørgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach saces, Ann. Probability 4 4), ). 5. D. H. Hong and S. Y. Hwang, Marcinkiewicz-tye strong law of large numbers for double arrays of airwise indeendent random variables, Int. J. Math. Math. Sci. 22 ), 7 999). 6. D. H. Hong, and A. I. Volodin, Marcinkewicz-tye law of large numbers for double arrays, J. Korean Math. Soc 36 6), ). 7. A. Gut and A. Sǎtaru, Precise asymtotics in some strong limit theorems for multidimensionally indexed random variables, J. Multivariate Anal. 86 2), ). 8. A. Gut, onvergence rates in the central limit theorem for multidimensionally indexed random variables, Studia Sci. Math. Hungar ), ). 9. S. Kwaień, andw. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multile Birkhäuser, Boston, 992).

10 346 DUNG et al. 0. G. Pisier, Martingales with values in uniformly convex saces, Israel J. Math ), ).. G. Pisier, Probabilistic methods in the geometry of Banach saces, in: Probability and Analysis Varenna, 985); Lecture Notes in Math. Sringer, Berlin, 986), Vol. 206, N. V. Quang, and L. V. Thanh, Marcinkiewicz-Zigmund law of large numbers for blockwise adated sequence, Bull. Korean Math. Soc. 43 ), A. Rosalsky and L. V. Thanh, Strong and weak laws of large numbers for double sums of indeendent random elements in Radermacher tye Banach saces, Stoch. Anal. Al. 24 6), ). 4. F. S. Scalora, Abstract martingale convergence theorems, PacificJ.Math., ). 5. L. V. Thanh, Mean convergence theorems and weak laws of large numbers for double arrays of random variables, J. Al. Math. Stoch. Anal. Art. ID ). 6. W. A. Woyczyśki,Geometry and martingales in Banach saces II. Indeendent increments. Probability on Banach saces, Adv. Probab. Related Toics Dekker, New York, 978), Vol. 4,. 267.