DSpace at VNU: MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES

Bull Korean Math Soc 47 (2010), No 3, pp 467–482 DOI 10.4134/BKMS.2010.47.3.467 MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES Le Van Dung and Nguyen Duy Tien Abstract For a double array of random elements {Vmn ; m ≥ 1, n ≥ 1} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established For the mean conver− Pum Pvn gence results, conditions are provided under which kmnr j=1 (Vij − i=1 E(Vij |Fij )) → in Lr (0 < r < 2) The weak law results provide con− PTm Pτn ditions for kmnr j=1 (Vij − E(Vij |Fij )) → in probability where i=1 {Tm ; m ≥ 1} and {τn ; n ≥ 1} are sequences of positive integer-valued random variables, {kmn ; m ≥ 1, n ≥ 1} is an array of positive integers The sharpness of the results is illustrated by examples Introduction The classical notion of uniform integrability of a sequence {Xn ; n ≥ 1} of integrable random variables is defined through the condition lim sup E|Xn |I(|Xn | > a) = a→∞ n≥1 Landers and Rogge [4] prove that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak law of large numbers Chandra [2] obtains the weak law of large numbers under the condition which is weaker than uniform integrability: the condition of Ces`aro uniform integrability A sequence {Xn ; n ≥ 1} of integrable random variables is said to be Ces` aro uniformly integrable if a→∞ n≥1 kn kn lim sup E|Xj |I(|Xj | > a) = 0, j=1 where {kn ; n ≥ 1} is a sequence of positive integers such that limn→∞ kn = ∞ Received August 8, 2008; Revised August 25, 2009 2000 Mathematics Subject Classification 60B11, 60B12, 60F15, 60F25, 60G42 Key words and phrases martingale type p Banach spaces, double arrays of random elements, weighted double sums, weak laws of large numbers, mean convergence theorem c 2010 The Korean Mathematical Society 467 468 LE VAN DUNG AND NGUYEN DUY TIEN Definition Let {Xn ; n ≥ 1} be a sequence of random variables and {anj ; ≤ j ≤ , n ≥ 1} be an array of constants with j=1 |anj | ≤ C for all n ∈ N and some constant C > The sequence {Xn ; n ≥ 1} is {anj }-uniform integrable if lim sup a→∞ n≥1 |anj |E|Xj |I(|Xj | > a) = 0, j=1 where {vn ; n ≥ 1} is a sequence of positive integers such that limn→∞ = ∞ Under the condition of {anj }-uniform integrability, Ord´on ˜ez Cabrera [5] obtains the weak law of large numbers for weighted sums of pairwise independent random variables; the condition of pairwise independence can be even dropped, at the price of slightly strengthening the conditions on the weights Recently, Thanh [10] obtains the mean convergence theorems for the weighted km ln sums i=1 j=1 amnij (Xij − EXij ) in Lp and the weak laws of large numT τ m n bers with random indices for the weighted sums i=1 j=1 amnij (Xij −EXij ), where {Xij , i ≥ 1, j ≥ 1} is an array of random variable, {amnij ; m, n, i, j ≥ 1} are constants, {Tm ; m ≥ 1} and {τn ; n ≥ 1} are sequences of positive integervalued random variables Sung [9] introduces the concept of Ces`aro type uniform integrability with exponent r Definition Let {Xn ; n ≥ 1} be a sequence of random variables and r > The array {Xn ; n ≥ 1} is said to be Ces` aro type uniformly integrable with exponent r if n≥1 kn a→∞ n≥1 kn E|Xj |r < ∞ and lim sup sup j=1 E|Xj |r I(|Xj | > a) = 0, j=1 where {kn ; n ≥ 1} and {vn ; n ≥ 1} are two sequences of positive integers such that limn→∞ kn = limn→∞ = ∞ In this paper, we not only enlarge on some results of Adler et al [1] and Thanh [10] but we also weaken the suppositions and bring more general results Preliminaries For a, b ∈ R, {a, b}, max {a, b} and the integer part of a will be denoted, respectively, by a ∧ b, a ∨ b and [a] Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance Technical definitions relevant to the current work will be discussed in this section Scalora [8] introduced the idea of the conditional expectation of a random element in a Banach space For a random element V and sub σalgebra G of F, the conditional expectation E(V |G) is defined analogously to that in the random variable case and enjoys similar properties MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 469 A real separable Banach space X is said to be martingale type p (1 ≤ p ≤ 2) if there exists a finite positive constant C such that for all martingales {Sn ; n ≥ 1} with values in X , ∞ p sup E Sn ≤C n≥1 E Sn − Sn−1 p n=1 It can be shown using classical methods from martingale theory that if X is of martingale type p, then for all ≤ r < ∞ there exists a finite constant C such that r ∞ E sup Sn r ≤ CE p p Sn − Sn−1 n≥1 n=1 Clearly every real separable Banach space is of martingale type and the real line (the same as any Hilbert space) is of martingale type If a real separable Banach space of martingale type p for some < p ≤ 2, then it is of martingale type r for all r ∈ [1, p) It follows from the Hoffmann-Jørgensen and Pisier [3] characterization of Rademacher type p Banach spaces that if a Banach space is of martingale type p, then it is of Rademacher type p But the notion of martingale type p is only superficially similar to that of Rademacher type p and has a geometric characterization in terms of smoothness For proofs and more details, the reader may refer to Pisier [6, 7] The following lemma is needed to prove our main results Lemma 2.1 Suppose that the array of random elements {Vmn ; m ≥ 1, n ≥ 1} is Ces` aro type uniform integrability with exponent r, in the sense that (2.1) um sup m≥1,n≥1 r E Vij kmn ≤M a) = 0, i=1 j=1 where {kmn ; m ≥ 1, n ≥ 1} is an array of positive integers such that lim m∨n→∞ kmn = ∞ Then um v n β r E Vij kmn β I( Vij r ≤ kmn ) → as m ∨ n → ∞ if r < β i=1 j=1 Proof β r kmn um v n E Vij i=1 j=1 β I( Vij r ≤ kmn ) 470 = ≤ = LE VAN DUNG AND NGUYEN DUY TIEN um kmn E Vij β r kmn i=1 j=1 l=1 um kmn l β r kmn i=1 j=1 l=1 um kmn l β r kmn ≤ β−r r I(l − < Vij r E Vij β−r r ≤ l) I(l − < Vij r (E Vij r I( Vij r r ≤ l) r > l − 1) − E Vij I( Vij r > l)) i=1 j=1 l=1 um v n E Vij β r kmn r r I( Vij > 0) i=1 j=1 + β um kmn −1 (l + 1) β r kmn i=1 j=1 =: Amn + Bmn β−r r − (l) β−r r E Vij r I( Vij r > l) l=1 For Amn we have Amn ≤ sup β r −1 m≥1,n≥1 kmn kmn by um E Vij r → as m ∨ n → ∞, i=1 j=1 β − > and (2.2) Next, we will show that r lim m∨n→∞ Bmn = For arbitrary ε > 0, since (2.3) there exists a0 such that sup m≥1,n≥1 kmn um E Vij r I( Vij r > a0 ) < i=1 j=1 Now for all large enough m ∨ n, we have kmn ≥ ([a0 ] + 1) ε r β−r 2M ε and Bmn = um [a0 ] (l + 1) β r kmn β r kmn ≤ β r kmn − (l) β−r r E Vij r I( Vij r > l) i=1 j=1 l=1 + β−r r um v n kmn −1 (l + 1) β−r r − (l) β−r r E Vij r I( Vij i=1 j=1 l=[a0 ]+1 um [a0 ] (l + 1) i=1 j=1 l=1 β−r r − (l) β−r r E Vij r I( Vij r > 1) r > l) MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS um v n + (l + 1) β r kmn β r ([a0 ] + 1) kmn + i=1 j=1 um v n β r (kmn ) kmn + − (l) β−r r r E Vij I( Vij r > [a0 ] + 1) β−r r β−r r r − E Vij − ([a0 ] + 1) I( Vij β−r r r > 1) r E Vij r I( Vij > [a0 ] + 1) i=1 j=1 β−r r [a0 ] + kmn ≤ β−r r i=1 j=1 l=[a0 ]+1 um = kmn −1 471 kmn m≥1,n≥1 kmn m≥1,n≥1 r E Vij i=1 j=1 um sup um v n sup E Vij r I( Vij r > [a0 ] + 1) < i=1 j=1 ε ε + = ε 2 Thus Bmn → as m ∨ n → ∞ The proof is completed To prove Lemma 2.3, we need the following lemma: Lemma 2.2 If {Xkl , Fl ; l ≥ 1}, k = 1, 2, , m are nonnegative submartingales, then {max1≤k≤m Xkl , Fl ; l ≥ 1} is a nonnegative submartingale Proof For L > l ≥ 1, E( max XkL |Fl ) ≥ max E(XkL |Fl ) ≥ max Xkl 1≤k≤m 1≤k≤m 1≤k≤m Let Fkl be the σ-field generated by the family of random elements {Vij ; i < k or j < l}, F1,1 = {∅; Ω} We have the following lemma: Lemma 2.3 Let {Vij ; ≤ i ≤ m, ≤ j ≤ n} be a collection of mn random elements in a martingale type p (1 ≤ p ≤ 2) Banach space with E(Vij |Fij ) = for all i, j (1 ≤ i ≤ m, ≤ j ≤ n) Then m (2.3) p n E Vij m i=1 j=1 P k l max  1≤k≤m 1≤l≤n E Vij p ; i=1 j=1   (2.4) n ≤C Vij i=1 j=1   C >ε ≤ p  ε m n E Vij p ∀ε > i=1 j=1 n k Proof First, we prove (2.3) Set Skn = i=1 j=1 Vij , since E(Vij |Fij ) = we have that {Skn , Gk = Fk+1,1 ; ≤ k ≤ m} is a martingale Thus m (2.5) p n E Vij i=1 j=1 m ≤ E max 1≤k≤m Skn p ≤C n E k=1 Vkj j=1 p 472 LE VAN DUNG AND NGUYEN DUY TIEN l j=1 On the other hand, for each k (1 ≤ k ≤ m), { is a martingale Hence, n (2.6) E Vkj , Gkl = Fk,l+1 ; ≤ l ≤ n} l Vkj j=1 p ≤ E max 1≤l≤n n Vkj p ≤C j=1 E Vkl p l=1 Combining (2.5) and (2.6) yields the conclusion (2.3) k l Next, we will prove (2.4) Set Skl = i=1 j=1 Vij , Yl = max1≤k≤m Skl If σl is a σ-field generated by {Vij ; ≤ i ≤ m, ≤ j ≤ l}, then for each l (1 ≤ l ≤ n), σl ⊂ Fi,l+1 for all i ≥ 1, which follows that E(Vi,l+1 |σl ) = E(E(Vi,l+1 |Fi,l+1 )|σl ) = Thus, we have k E(Sk,l+1 |σl ) = E(Skl |σl ) + E(Vi,l+1 |σl ) = Skl i=1 It means that {Skl , σl ; ≤ l ≤ n} is a martingale Hence, { Skl , σl ; ≤ l ≤ n} is a nonnegative submartingale for each k = 1, 2, , m, which follows by Lemma 2.1 that {Yl , σl ; ≤ l ≤ n} is a nonnegative submartingale Applying Kolmogorov’s inequality, we obtain   k l   max P Vij > ε = P { max Yl > ε} ≤ p EYnp 1≤k≤m  1≤l≤n  1≤l≤n ε i=1 j=1 = ≤ ≤ E max Skn εp 1≤k≤m C εp C εp C = p ε m p ≤ C εp m n E k=1 Vkj p j=1 l E max 1≤l≤n k=1 m n Vkj p j=1 E Vkl p E Vij p k=1 l=1 m n i=1 j=1 The proof is completed Main results With the preliminaries accounted for, the main results may now be established In the following we let {Vmn ; m ≥ 1, n ≥ 1} be an array of random elements defined on a probability (Ω, F, P ) and taking values in a real separable Banach space X with norm · , Fkl be a σ-field generated by {Vij ; i < k or j < l}, F1,1 = {∅; Ω} Let {un ; n ≥ 1}, {vn ; n ≥ 1} be sequences of positive integers such that limn→∞ un = limn→∞ = ∞, {kmn ; m ≥ 1, n ≥ 1} be an array of positive integers such that limm∨n→∞ kmn = ∞ MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 473 Theorem 3.1 Let ≤ r < p ≤ 2, Banach space X be a martingale type p Suppose that {Vmn ; m ≥ 1, n ≥ 1} satisfies the Ces` aro type uniform integrability with exponent r, in the sense that (2.1) and (2.2) hold Then um (Vij − E(Vij |Fij )) → in Lr as n ∨ m → ∞ r kmn i=1 j=1 Proof For each m, n, ≤ i ≤ um , ≤ j ≤ , set Vmnij = Vij I( Vij r ≤ kmn ), Vmnij = Vij I( Vij r > kmn ), Umnij = E(Vmnij |Fij ), Umnij = E(Vmnij |Fij ) Observe that for each i and j, ≤ i ≤ um , ≤ j ≤ , we have Vij − E(Vij |Fij ) = (Vmnij − Umnij ) + (Vmnij − Umnij ) and E(Vmnij − Umnij |Fij ) = E(Vmnij − Umnij |Fij ) = Hence, E i=1 j=1 um kmn ≤C Vij r kmn = E (Vmnij − Umnij ) + E r (Vmnij − Umnij ) i=1 j=1 kmn kmn  r um v n E (Vmnij − Umnij ) (by cr -inequality) i=1 j=1   ≤C E kmn +C (Vmnij − Umnij ) i=1 j=1 um kmn r um i=1 j=1 +C ≤C r um p r/p um (Vmnij − Umnij )  i=1 j=1 r um v n E (Vmnij − Umnij ) (by Liapunov’s inequality) i=1 j=1 r/p um  E (Vmnij − Umnij ) p  kmn i=1 j=1 +C kmn um v n E (Vmnij − Umnij ) i=1 j=1 r (by Lemma 2.3) 474 LE VAN DUNG AND NGUYEN DUY TIEN  v u  m n ≤C E Vij kmn i=1 j=1 +C ≤ kmn ) r I( Vij r E Vij kmn ≤ C p r kmn  p um v n  r/p I( Vij r > kmn ) i=1 j=1 (by cr -inequality) r/p um E Vij p I( Vij r ≤ kmn ) i=1 j=1 + C  sup m≥1,n≥1  um kmn E Vij r r I( Vij > kmn ) → as m ∨ n → ∞, i=1 j=1 by Lemma 2.1 with β = p and (2.2) The proof is completed Corollary 3.2 Let ≤ r < p ≤ 2, Banach space X be a martingale type p Let {Amnij ; m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ } be an array of random um r variables satisfying i=1 j=1 E|Amnij | ≤ C < ∞ for all m ≥ 1, n ≥ Suppose that {Amnij ; ≤ i ≤ um , ≤ j ≤ } and {Vij ; i ≥ 1, j ≥ 1} are independent for all m ≥ 1, n ≥ Assume that the following conditions hold: (i) { Vmn r ; m ≥ 1, n ≥ 1} is {E|Amnij |r }-uniformly integrable, in the sense that um v n lim sup a→∞ m≥1,n≥1 (ii) Then sup 1≤i≤um ,1≤j≤vn E Amnij Vij r I( Vij > a) = 0; i=1 j=1 E|Amnij | → as m ∨ n → ∞ um Amnij (Vij − E(Vij |Fij )) → in Lr as m ∨ n → ∞ i=1 j=1 Proof Let   kmn =   sup 1≤i≤um ,1≤j≤vn E|Amnij |   Then kmn → ∞ as m ∨ n → ∞ It is easy to prove that um sup m≥1,n≥1 i=1 j=1 E Amnij Vij r < ∞ 1/r Hence, take kmn Amnij Vij instead Vij in Theorem 3.1, we have by (i) that sup n≥1,m≥1 kmn um v n um 1/r E kmn Amnij Vij i=1 j=1 r = sup m≥1,n≥1 i=1 j=1 E Amnij Vij r < ∞ MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 475 On the other hand, for all large enough m ∨ n we have sup 1≤i≤um ,1≤j≤vn E|Amnij | ≤ 1, which follows that E|Amnij |r ≤ sup E|Amnij | kmn E|Amnij |r ≤ E|Amnij |r ≤ sup E|Amnij |r 1≤i≤um ,1≤j≤vn 1≤i≤um ,1≤j≤vn Therefore, for all large enough m ∨ n, kmn |Amnij |r ≤ a.s Thus, it follows by (i) that lim um sup a→∞ m≥1,n≥1 kmn um ≤ lim 1/r Amnij Vij E kmn I( Vij r > a) i=1 j=1 sup a→∞ m≥1,n≥1 r E Amnij Vij r I( Vij > a) = i=1 j=1 This implies lim um sup a→∞ m≥1,n≥1 kmn 1/r Amnij Vij E kmn r I( Vij r > a) = i=1 j=1 By Theorem 3.1 we obtain the proof Theorem 3.3 Let < r < Suppose that {Vmn ; m ≥ 1, n ≥ 1} satisfies the Ces` aro type uniform integrability with exponent r, in the sense that (2.1) and (2.2) hold Then 1 r kmn um Vij → in Lr as n ∨ m → ∞ i=1 j=1 Proof E = ≤ ≤ Vij r kmn kmn kmn = i=1 j=1 E kmn E Vij i=1 j=1 r um Vij I( Vij r ≤ kmn ) + i=1 j=1 Vij I( Vij > kmn ) i=1 j=1 r um v n E r um um v n Vij I( Vij r ≤ kmn ) + um Vij I( Vij i=1 j=1 kmn r i=1 j=1   E kmn r um v n r ≤ kmn )  r um E Vij I( Vij i=1 j=1 r > kmn ) 476 LE VAN DUNG AND NGUYEN DUY TIEN + kmn  ≤  r kmn  r um E Vij I( Vij r > kmn ) i=1 j=1 (by Liapunov’s inequality) r um E Vij I( Vij r ≤ kmn ) i=1 j=1 +  sup m≥1,n≥1  um kmn E Vij r I( Vij r > kmn ) → as m ∨ n → ∞, i=1 j=1 by Lemma 2.1 with β = and (2.2) The proof is completed Corollary 3.4 Let < r < and {Amnij ; m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ um r } be an array of random variables satisfying i=1 j=1 E|Amnij | ≤ C < ∞ for all m ≥ 1, n ≥ Suppose that Amnij and Vij are independent for all m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ Assume that the following conditions hold: (i) { Vmn r ; m ≥ 1, n ≥ 1} is {E|Amnij |r }-uniformly integrable; (ii) sup E|Amnij | → as m ∨ n → ∞ Then 1≤i≤um ,1≤j≤vn um Amnij Vij → in Lr as m ∨ n → ∞ i=1 j=1 Proof The proof is similar to that of Corollary 3.2 In the following theorem, we establish the weak law of large numbers with random indices for weighted double sums of random elements Theorem 3.5 Let ≤ r < p ≤ 2, Banach space X be a martingale type p Suppose that {Vmn ; m ≥ 1, n ≥ 1} satisfies the Ces` aro type uniform integrability with exponent r, in the sense that (2.1) and (2.2) hold Let {Tn ; n ≥ 1} and {τn ; n ≥ 1} be sequences of positive integer-valued random variables such that (3.1) lim P {Tn > un } = lim P {τn > } = n→∞ n→∞ Then (3.2) Tm τn 1 r kmn P (Vij − E(Vij |Fij )) −→ as n ∧ m → ∞ i=1 j=1 Proof For arbitrary ε > 0,   Tm τn   (3.3) P (V − E(V |F )) > ε ij ij ij  k r1  mn i=1 j=1 MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS   ≤P   kr  Tm τn (Vij − E(Vij |Fij )) > ε (Tm ≤ um ) i=1 j=1 mn 477   (τn ≤ )  + P {Tm > um } + P {τn > } Set the same Vmnij , Vmnij , Umnij and Umnij as in the proof of Theorem 3.1 we have    Tm τn   (3.4) P  (Vij − E(Vij |Fij )) > ε (Tm ≤ um ) (τn ≤ )  kr  mn i=1 j=1    k l   um v n  11 ≤P (Vij − E(Vij |Fij )) > ε   r kmn i=1 j=1 i=1 j=1   k l   =P max max (V − E(V |F )) > ε ij ij ij  k r1 1≤k≤um 1≤l≤vn  i=1 j=1 mn   k l  ε ≤P max max (V − U ) > mnij mnij  k r1 1≤k≤um 1≤l≤vn 2 i=1 j=1 mn   k l  ε +P max max (Vmnij − Umnij ) >  k r1 1≤k≤um 1≤l≤vn 2 i=1 j=1 mn ≤ um C + C εr k mn i=1 j=1  ≤ p E Vmnij − Umnij p r εp kmn i=1 j=1 um C p r εp kmn E Vmnij − Umnij E Vij i=1 j=1  um v n (by Lemma 2.3)  um  r p I( Vij r ≤ kmn )  C  E Vij r I( Vij r > kmn ) (by cr -inequality) εr kmn i=1 j=1   um C E Vij p I( Vij r ≤ kmn ) ≤ p p r ε kmn i=1 j=1   um v n C + r  sup E Vij r I( Vij r > kmn ) → ε m≥1,n≥1 kmn i=1 j=1 + as m ∧ n → ∞, 478 LE VAN DUNG AND NGUYEN DUY TIEN by Lemma 2.1 with β = p and (2.2) The conclusion (3.2) follows immediately from (3.1), (3.3) and (3.4) The below corollary is inferred from the above theorem and the proof is similar to that of Corollary 3.2 Corollary 3.6 Let ≤ r < p ≤ 2, Banach space X be a martingale type p Let {Amnij ; m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ } be an array of random variables um r satisfying i=1 j=1 E|Amnij | ≤ C < ∞ for all m ≥ 1, n ≥ Suppose that {Amnij ; ≤ i ≤ um , ≤ j ≤ } and {Vij ; i ≥ 1, j ≥ 1} are independent for all m ≥ 1, n ≥ Let {Tn ; n ≥ 1} and {τn ; n ≥ 1} be sequences of positive integervalued random variables satisfying (3.1) Assume that the following conditions hold: (i) { Vmn r ; m ≥ 1, n ≥ 1} is {E|Amnij |r }-uniformly integrable; (ii) sup E|Amnij | → as m ∨ n → ∞ Then 1≤i≤um ,1≤j≤vn Tm τn P Amnij (Vij − E(Vij |Fij )) −→ as n ∧ m → ∞ i=1 j=1 Remark 3.7 In the case of r = p, the conclusion of Corollaries 3.2 and 3.6 will fail (see Example 4.1 in Section 4) However, if the condition (ii) is replaced um p by stronger condition: i=1 j=1 E|Amnij | = as m ∨ n → ∞, then the conclusion of Corollary 3.1 is still right when the hypothesis that ≤ r < p ≤ is replaced by ≤ r ≤ p ≤ In the case of < r < we also have the result which is similar to Theorem 3.5 Theorem 3.8 Let < r < Suppose that {Vmn ; m ≥ 1, n ≥ 1} satisfies the Ces` aro type uniform integrability with exponent r, in the sense that (2.1) and (2.2) hold Let {Tn ; n ≥ 1} and {τn ; n ≥ 1} be sequences of positive integervalued random variables satisfying (3.1) Then 1 r kmn Tm τn P Vij −→ as n ∧ m → ∞ i=1 j=1 Corollary 3.9 Let < r < Let {Amnij ; m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ um r } be an array of random variables satisfying i=1 j=1 E|Amnij | ≤ C < ∞ for all m ≥ 1, n ≥ Suppose that Amnij and Vij are independent for all m ≥ 1, n ≥ 1, ≤ i ≤ um , ≤ j ≤ Let {Tn ; n ≥ 1} and {τn ; n ≥ 1} be sequences of positive integer-valued random variables satisfying (3.1) Assume that the following conditions hold: (i) { Vmn r ; m ≥ 1, n ≥ 1} is {E|Amnij |r }-uniformly integrable; (ii) sup E|Amnij | → as m ∨ n → ∞ 1≤i≤um ,1≤j≤vn MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 479 Then Tm τn P Amnij Vij −→ as m ∨ n → ∞ i=1 j=1 Some interesting examples Four illustrative examples will now be presented For ≤ q ≤ 2, let lq denote the Banach space of absolute q th power summable real sequence v = {vi , i ≥ 1} ∞ with norm v = ( i=1 |vi |q )1/q The element having α (α ∈ R) in its k th (k) position and elsewhere will be denoted by vα , k ≥ Let ϕ : N × N → N be one-to-one and onto map, where N = {1, 2, 3, } is the set of all positive integers The first example shows that the hypothesis that ≤ r < p ≤ in Theorem 3.1 can not be replaced by the hypothesis that ≤ r ≤ p ≤ Example 4.1 Let r = p = and consider the Banach space l2 and the array of independent random elements {Vij ; i ≥ 1, j ≥ 1} in l2 with (ϕ(i,j)) (ϕ(i,j)) P Vij = v1 = P Vij = −v1 = , i, j ≥ Then Vij = a.s., therefore, (2.2) is automatic Suppose that um = m, = n and kmn = mn Then (2.1) holds since um v n sup m≥1,n≥1 kmn E Vij = i=1 j=1 Moreover, with probability 1 kmn um Vij = i=1 j=1 Thus, the conclusion of Theorem 3.1 fails The next example illustrates the essential role that the condition (2.1) plays in Theorem Example 4.2 Let 3/2 < p ≤ 2, r = 2p/3, un = = n2 , kmn = (mn)β with β ∈ N and consider the Banach space lp and the array of independent random elements {Vij ; i ≥ 1, j ≥ 1} in lp with (ϕ(i,j)) (ϕ(i,j)) P Vij = v1 = P Vij = −v1 = , i, j ≥ Then Vij = a.s., therefore, (2.2) is automatic If β ≥ 2, then (2.1) holds since sup m≥1,n≥1 kmn um E Vij i=1 j=1 r = sup (mn)2−β ≤ m≥1,n≥1 480 LE VAN DUNG AND NGUYEN DUY TIEN Hence, um Vij → in Lr as m ∨ n → ∞ r kmn i=1 j=1 On the other hand, if β = 1, then um v n sup E Vij m≥1,n≥1 kmn r = sup (mn) = ∞, m≥1,n≥1 i=1 j=1 whence (2.1) fails Moreover, with probability r um kmn 2r = (mn) p −1 = (mn) ≥ 1, Vij i=1 j=1 and so the conclusion of Theorem 3.1 fails The third example shows that in Theorems 3.1 the condition (2.2) cannot be dispensed with Example 4.3 Let r = 1, p = 2, un = = n, kmn = 2m 2n and consider the Banach space l2 and the array of independent random elements {Vij ; i ≥ 1, j ≥ 1} in l2 with P Vij = vα(ϕ(i,j)) = P Vij = −vα(ϕ(i,j)) = , i, j ≥ 1, ij ij where αij = 2(i−1) 2(j−1) Then Vij = 2(i−1) 2(j−1) a.s Therefore, (2.1) holds since sup m≥1,n≥1 kmn um E Vij r = i=1 j=1 m n m≥1,n≥1 2 m n 2(i−1) 2(j−1) sup m = sup m≥1,n≥1 i=1 j=1 n (2 − 1)(2 − 1) ≤ 2m 2n However, (2.2) fails since with a = 2(m+n−4) → ∞ as m ∨ n → ∞ but kmn um v n E Vij r I( Vij r > 2m+n−4 ) = i=1 j=1 Moreover, with probability 1 kmn r um Vij i=1 j=1 2m−1 2n−1 = → 2m 2n 1/2  m n  4i−1 4j−1  = m n 2 i=1 j=1 = m 2n = (4m − 1)(4n − 1) (4m − 1)(4n − 1) 4m 4n 1/2 1/2 ≥ , MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 481 so the conclusion of Theorem 3.1 fails Apropos of Theorem 3.3, the last example shows that its hypotheses not guarantee that the convergence in mean of order r prevails in the conclusion (3.2) Example 4.4 Let ≤ r < p ≤ and consider the Banach space lp and the array of independent random elements {Vij ; i ≥ 1, j ≥ 1} in lp with , i, j ≥ Suppose that un = = 2n , kmn = 2m 2n Then, it is easy to see that (2.1) and (2.2) hold Let {Tn , n ≥ 1} and {τn , n ≥ 1} be sequences of identically distributed random variables with the distribution of T1 and τ1 given by (ϕ(i,j)) P Vij = v1 (ϕ(i,j)) = P Vij = −v1 = P {T1 = 2i } = P {τ1 = 2i } = 2−i and suppose that {Tn ; 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