VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 Weak Laws of Large Numbers of Cesaro Summation for Random Arrays Tran Manh Cuong*, Ta Cong Son VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam Received 09 April 2015 Revised 10 June 2015; Accepted 06 August 2015 Abstract: In this paper, we establish weak laws of large numbers with or without random indices for Cesaro summation for random arrays of random elements in Banach spaces Our results are more general and stronger than some well-known ones AMS Subject classification 2000: 60B11, 60B12, 60F05, 60G42 Keywords: p-uniformly smooth Banach space, double arrays of random elements, double arrays, random indices, weak laws of large numbers Introduction∗ Consider a double array { X mn ; m ≥ 1, n ≥ 1} of random elements defined on a probability ‖‖ space (Ω, F , P ) taking values in a real separable Banach space E with norm Let {un ; n ≥ 1} and {vn ; n ≥ 1} be sequences of positive integers, let {Tn ; n ≥ 1} and {τ n ; n ≥ 1} be sequences of positive integer-valued random variables In the current work, we extend weak laws of large numbers of Cesaro summation for random arrays and for double arrays with random indices Limit theorems for weighted sums (with or without random indices) for random variables (realvalued or Banach space-valued) are studied by many authors (see, e.g., Wei and Taylor [1], OrdonezCabrera [2], Adler et al [3], Sung et al [4]) Recently, Dung [5] obtained the weak law of largenumbers with random indices for double arrays of random elements In this paper, we establish theweak laws of large numbers with or without random indices for Cesaro summation for random arrays ofrandom elements in a p- uniformly smooth Banach space Preliminaries For α > −1 , we let Anα = (α + 1)(α + 2) (α + n) , n = 1,2,3, and A0α = n! _ ∗ Corresponding author Tel.: 84-4-38581135 Email: cuongtm@vnu.edu.vn 31 T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 32 Then the definition of Cesaro summability for array is extended as follows Definition Let α , β > The array {x mn ; m, n ≥ 0} is said to be (C , α , β ) -summable iff α β Am An m ,n ∑ Aα −1 n −k Anβ−−l1 xkl converges as m, n → ∞ k ,l = It is easy to see that (C, 1, 1)-convergence is the same as convergence of array of the arithmetic mean Here we collect some facts that will be used on and off in general without specific reference Firstly, we have Anα ~ nα as n → ∞ Γ(α + 1) (1.1) Secondly, we use the fact that if {ak , k ≥ 1} is a sequence of numbers such that An ∞ as n n → ∞ where An = ∑ ak and xn → as n → ∞ then k =1 An n ∑a x k k → as n → ∞ (1.2) k =1 For a, b ∈ » , min{a, b} and max{a, b} will be denoted, respectively, by a ∧ b, a ∨ b 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 that are relevant to the current work will be discussed in this section Scalora [6] introduced the idea of the conditional expectation of a random element in a Banach space For a random element X and sub σ–algebra G of F , the conditional expectation E ( X | G ) is defined analogously to that in the random variable case and enjoys similar properties A real separable Banach space E is said to be p-uniformly smooth (1 ≤ p ≤ 2) if there exists a finite positive constant C such that for all martingales {S n ; n ≥ 1} with values in E , ‖ ‖ ≤ C ∑ E‖S ∞ sup E S n n ≥1 p n =1 n ‖ p − S n −1 It can be shown by using classical methods from martingale theory that if E is p-uniformly smooth, then for each ≤ r < ∞ there exists a finite constant C such that ‖‖ ‖ ‖  ∞ E sup S n r ≤ CE  ∑ Sn − S n −1 n ≥1  n =1 r p p    Clearly, every real separable Banach space is 1-uniformly smooth and the real line (the same as any Hilbert space) is 2-uniformly smooth It follows from the Hoffmann-Jøgensen and Pisier [7] that if a Banach space is p-uniformly smooth, then it is of Rademacher type p But the notion of p-uniformly smooth is only superficially similar to that of Rademacher type p and has a geometric characterization in terms of smoothness T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 33 Let F kl be the σ-field generated by the family of random elements { X ij ; i < k or j < l} , F 1,1 = {∅; Ω} The following lemma which is due to Dung [5] establishes a maximal inequality for double sums of random elements in martingale type p Banach spaces Lemma Let < p ≤ Let { X ij ;1 ≤ i ≤ m,1 ≤ j ≤ n} be a collection of mn random elements in a real separable Banach space such that E ( X ij | Fkl ) = for all ≤ i ≤ m,1 ≤ j ≤ n Then, k E max 1≤ k ≤ m 1≤ l ≤ n p l ∑∑ X ij m n ‖ ‖ ≤ C ∑∑ E X ij i =1 j =1 i =1 j =1 p (1.3) where the constant C is independent of m and n Random elements { X mn ; m ≥ 1, n ≥ 1} are said to be stochastically dominated by a random element X if for some finite constant D ‖ ‖ ‖ ‖ P{ X mn > t} ≤ DP{ DX > t}, t ≥ 0, m ≥ 1, n ≥ The main results Let { X mn ; m ≥ 1, n ≥ 1} be an array of random elements defined on a probability space (Ω, F , P ) ‖‖ and taking values in a real separable Banach space E with norm , F kl be a σ-field generated by { X ij ; i < k or j < l} , F 1,1 = {∅; Ω} Let {un ; n ≥ 1} , {vn ; n ≥ 1} be sequences of positive integers such that lim un = lim = ∞ For any set A, we denote I (A) the indicator function, i.e, n →∞ n →∞ 1 I ( A)(ω ) =  0 if ω ∈ A if ω ∉ A Set ‖ ‖ Yklmn = Amα −−1k Anβ−−l1 X kl I ( Amα −−1k Anβ−−l1 X kl ≤ Amα Anβ ) where α,β > Theorem Let ≤ p ≤ , α , β > and E be a p-uniformly smooth Banach space Suppose that um ∑∑ P{ Aα −1 m −i ‖ ‖ Anβ−−j1 X ij > Amα Anβ } → as m ∨ n → ∞ i =1 j =1 (2.1) and ∑∑ E‖Y um α β p ( Am An ) mn ij i =1 j =1 ‖ → as m ∨ n → ∞ − E (Yijmn | Gij ) p (2.2) Then 1≤ k ≤ um A Aβ m n max 1≤ l ≤ α k l ∑∑ Aα −1 m −i i =1 j =1 p Anβ−−j1 X ij − E (Yijmn | G ij ) → as m ∨ n → ∞ (2.3) T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 34 Proof For an arbitrary ε > 0,   P  max α β 1≤ k ≤ u  1≤l ≤ vnm Am An   α −1 β −1 mn A A X − E Y G > ε ( | )  ∑∑ m −i n − j ij ij ij i =1 j =1  k  ≤ P  max α β 1≤ k ≤ u  1≤l ≤ vnm Am An  α −1 β −1 mn A A X − Y > ε /  ∑∑ m −i n − j ij ij i =1 j =1  k   + P  max α β 1≤ k ≤ um A A  1≤ l ≤ m n u  m l l k l ∑∑ (Y mn ij mn ij − E (Y i =1 j =1   | G ij )) > ε /    ≤ P ∪∪ ( Amα −−i1 Anβ−−j1‖X ij‖> Amα Anβ )   i =1 j =1    + P  max α β 1≤ k ≤ um A A  1≤ l ≤ m n um   mn mn ( Y − E ( Y | G )) > ε /  ∑∑ ij ij ij i =1 j =1  k l ‖ ‖ >A A ) ≤ ∑∑ P ( Amα −−i1 Anβ−−j1 X ij i =1 j =1 2p + p α β p E max ε ( Am An ) 11≤≤kl ≤≤uvm n um k β n p l (Y ∑∑ i =1 j =1 mn ij mn ij − E (Y | Gij )) (by Markov’s inequality) ‖ ‖ >A A ) ≤ ∑∑ P ( Amα −−1k Anβ−−l1 X ij i =1 j =1 + α m α β m n um C mn mn p α β p ∑∑ E‖Yij − E (Yij ‖G ij )‖ (by Lemma 2) ( Am An ) i =1 j =1 → as m ∨ n → ∞ (by (3.1) and (3.2)) The proof is completed Corollary Let ≤ p ≤ 2, α > 0, β > and E be a p-uniformly smooth Banach space If um ∑∑ P{ Aα −1 m−i ‖ ‖ Anβ−−j1 X ij > Amα Anβ } → ∞ as m ∨ n → ∞, i =1 j =1 1≤ k ≤ um A Aβ m n max 1≤ l ≤ and α k l ∑∑ Aα −1 m −i i =1 j =1 p Anβ−−j1 E (Yijmn | G ij ) →0 as m ∨ n → ∞ T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 α β p ( Am An ) ∑∑‖E (Y | G um ‖→0 mn ij kl ) i =1 j =1 p 35 (2.4) as m ∨ n → ∞, then max 1≤ k ≤ um 1≤ l ≤ k α β Am An l p ∑∑ Amα −−i1 Anβ−−j1 X ij → i =1 j =1 (2.5) as m ∨ n → ∞ Remark If the condition (3.4) is replaced by the condition that α β Am An um p ∑∑ Amα −−1i Anβ−−j1E (Yijmn | Gij ) →0 as m ∨ n → ∞, i =1 j =1 then the conclusion (3.5) will be replaced by α β Am An k l ∑∑ Aα −1 m −i p Anβ−−j1 X ij → as m ∨ n → ∞ i =1 j =1 The following result is a random index version of Theorem Theorem Let ≤ p ≤ 2, α , β > and E be a p-uniformly smooth Banach space Suppose that {Tn ; n ≥ 1} and {τ n ; n ≥ 1} are sequences of positive integer-valued random variables such that (2.6) lim P{Tn > un } = lim P{τ n > } = n →∞ n →∞ If um P{ Amα −−i1 Anβ−−j1‖X ij‖> Amα Anβ } → ∞ as m ∨ n → ∞ ∑∑ i =1 j =1 and um mn mn p α β p ∑∑ E‖Yij − E (Yij | Gij )‖ → as m ∨ n → ∞, ( Am An ) i =1 j =1 then 1≤ k ≤Tm Aα Aβ m n max 1≤ l ≤τ n k l ∑∑ Aα −1 m−i P Anβ−−j1 ( X ij − E (Yijmn | G ij )) → as m ∨ n → ∞ i =1 j =1 Proof For arbitrary ε > 0, T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 36   k α 1≤ k ≤Tm A Aβ m n    l ∑∑ Amα −−1i Anβ−−j1 ( X ij − E (Yijmn | Gij )) > ε  P  max   i =1 j =1  1≤l ≤τ n   ≤ P  max α β  1≤ k ≤Tm Am An  1≤l ≤τ n  + P (Tm > um ) + P (τ n > )   ≤ P  max α β 1≤ k ≤ um A A   1≤l ≤ m n  k k l Am −i An − j ( X ij − E (Y ∑∑ i =1 j =1 l ∑∑ A α −1 m −i α −1 β −1 β −1 mn ij mn ij An − j ( X ij − E (Y i =1 j =1     | Gij )) > ε ∩ (Tm ≤ um ) ∩ (τ n ≤ )        | G ij )) > ε    + P (Tm > um ) + P (τ n > ) → as m ∨ n → ∞, (by (3.6) and Theorem 3) which completes the proof We shall now prove the following extension of the well-known Feller theorem for Cesaro summation for random arrays of random elements in Banach spaces Theorem Let ≤ p ≤ 2, < α < β ≤ 1, E be a p-uniformly smooth Banach space Suppose that { X mn ; m ≥ 1, n ≥ 1} is stochastically dominated by a random element X If ‖‖ (2.7) lim nP{ X > n} = 0, n →∞ then 1≤ k ≤ m Aα A β m n max k l ∑∑ Aα 1≤ l ≤ n −1 m −i P Anβ−−j1 ( X ij − E (Yijmn | Gij )) →0 as m ∨ n → ∞ i =1 j =1 Proof We verify (3.1) and (3.2) with um = m , = n For (3.1), we have m ,n ∑ P{ A α −1 m −i ‖ ‖ m,n Anβ−−j1 X ij > Amα Anβ } ≤ C ∑ P{ Amα −−1i Anβ−−j j i , j =1 i , j =1 ‖X‖> A A } α β m n ‖ ‖ m ,n ≤ C ∑ P{i α −1 j β −1 X ij > mα n β } by (2.1) i , j =1 =C mα n β m,n ∑i α −1 ‖ ‖ j β −1 mα m β i1−α j1− β P{i α −1 j β −1 X ij > mα n β } → i , j =1 (by (2.2) and (3.7)) For (3.2), by Jensen’s inequality for conditional expectation, we get m n m n C mn mn p E ‖ Y − E ( Y | G ) ‖ ≤ E‖Yijmn‖p ∑∑ ij ij ij ∑∑ α β p α β p ( Am An ) i =1 j =1 ( Am An ) i =1 j =1 T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 ≤ 37 m n C E (i p (α −1) j p ( β −1) ‖ | X ij‖p I (iα −1 j β −1‖X ij‖≤ mα n β )) mα p n β p ∑∑ i =1 j =1 β /α m n mn C = α p β p ∑∑ i p (α −1) j p ( β −1) ∑ ‖X ij‖p I ((k − 1)α < iα −1 j β −1‖X ij‖≤ k α ) m n i =1 j =1 k =1 β /α m n mn C ≤ α p β p ∑∑ ∑ k pα P((k − 1)α < iα −1 j β −1‖X‖≤ k α ) m n i =1 j =1 k =1 β /α = m n mn C pα ( P(iα −1 j β −1‖X‖> (k − 1)α ) − P (iα −1 j β −1‖X‖> k α )) α p β p ∑∑ ∑ k m n i =1 j =1 k =1 β /α m n mn C ≤ α p β p ∑∑ ∑ k pα −1 P (iα −1 j β −1‖X‖≥ k α ) m n i =1 j =1 k =1 C ≤ α β m n m n i ∑∑ i =1 j =1 α −1 β −1 j pα −α pα −α m n mn β /α k pα −α −1 (k α i1−α j1− β P‖ ( X‖≥ k α i1−α j1− β )) ∑ k =1 → as m, n → ∞ By applying Theorem 3, the proof is completed The following result is a random index version of Theorem Theorem Let ≤ p ≤ 2, < α , β ≤ 1, E be a p-uniformly smooth Banach space Suppose that { X mn ; m ≥ 1, n ≥ 1} is stochastically dominated by a random element X Suppose that {Tn ; n ≥ 1} and {τ n ; n ≥ 1} are sequences of positive integer-valued random variables such that (3.6) lim P{Tn > n} = lim P{τ n > n} = n →∞ If n →∞ ‖‖ lim nP X > n = 0, n →∞ then 1≤ k ≤Tm A Aβ m n max 1≤ l ≤τ n α k l ∑∑ Aα −1 m−i P Anβ−−j1 ( X ij − E (Yijmn | G ij )) →0 as m ∨ n → ∞ i =1 j =1 Proof By the same argument in the proof of Theorem and using Theorem Acknowledgement This work was partially supported by the VNU research project No QG.13.02 38 T.M Cuong, T.C Son / VNU Journal of Science: Mathematics – Physics, Vol 31, No (2015) 31-38 References [1] D Wei and R L Taylor, Convergence of weighted sums of tight random elements, Journal of Multivariate Analysis 8(1978), no 2, 282-294 [2] M Ordonez Cabrera, Limit theorems for randomly weighted sums of random elements in normed linear spaces, Journalof Multivariate Analysis 25 (1988), no 1, 139-145 [3] A Adler, A Rosalsky and R L Taylor, A weak law for normed weighted sums of random elements in Rademacher typep Banach spaces, Journal of Multivariate Analysis 37 (1991), no 2, 259-268 [4] S H Sung, T.C Hu and A.I Volodin, On the weak laws with random indices for partial sums for arrays of randomelements in martingale type p Banach spaces, Bull.Korean.Soc 43 (2006), no 3, 543-549 [5] L.V Dung, Weak law of large numbers for double arrays of random elements in Banach spaces, Acta MathematicaVietnamica 35 (2010), 387-398 [6] F S Scalora, Abstract martingale convergence theorems, Pacific J Math 11 (1961), 347-374 [7] J Hoffmann-Jorgensen and G Pisier, The law of large numbers and the central limit theorem in Banach spaces, Annalsof Probability (1976), no 4, 587-599  ... partial sums for arrays of randomelements in martingale type p Banach spaces, Bull.Korean.Soc 43 (2006), no 3, 543-549 [5] L.V Dung, Weak law of large numbers for double arrays of random elements... and Theorem 3) which completes the proof We shall now prove the following extension of the well-known Feller theorem for Cesaro summation for random arrays of random elements in Banach spaces Theorem... theorems for randomly weighted sums of random elements in normed linear spaces, Journalof Multivariate Analysis 25 (1988), no 1, 139-145 [3] A Adler, A Rosalsky and R L Taylor, A weak law for normed