ON THE STRONG LIMIT THEOREMS FOR DOUBLE ARRAYS OF BLOCKWISE M-DEPENDENT RANDOM VARIABLES

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1 Available at: IC/28/89 United Nations Educational, Scientific Cultural Organization International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON THE STRONG LIMIT THEOREMS FOR DOUBLE ARRAYS OF BLOCKWISE M-DEPENDENT RANDOM VARIABLES Ulrich Stadtmüller Department of Number Probability Theory, Ulm University, 8969 Ulm, Germany Le Van Thanh 2 Department of Mathematics, Vinh University, Vinh, Vietnam The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract For a double array of blockwise M-dependent rom variables {X mn,m,n }, strong laws of large numbers are established for double sum m i= n j= X ij, m,n. The main results are obtained for (i) the rom variables {X mn,m,n } are non-identically distributed (ii) the rom variables {X mn,m,n } are identically distributed. MIRAMARE TRIESTE December 28 ulrich.stadtmueller@uni-ulm.de 2 levt@vinhuni.edu.vn

2 . Introduction Móricz [5] introduced the concept of blockwise independence for a sequence of rom variables extended a classical strong law of large numbers (SLLN) of Kolmogorov (see, e.g., Chow & Teicher, [], p. 24) to the blockwise independent case. Móricz s [5] result was extended by Gaposhkin [2]. The SLLN for double arrays of blockwise independent rom variables was also studied by Quang Thanh [7]. Recently, Móricz, Stadtmuller Thalmaier [6] introduced the concept of blockwise M-dependence for a double array of rom variable established a double array version of the Kolmogorov SLLN for double arrays of rom variables which are blockwise M- dependent with respect to the blocks {[2 k,2 k+ ) [2 l,2 l+ ),k,l }. In this paper, besides studying this problem for double arrays of rom variables which are blockwise M-dependent with respect to arbitrary blocks, we also consider the Marcinkiewicz-Zygmund SLLN. When the rom variables are non-identically distributed, we establish a new type of SLLN, it is different from the result of Quang Thanh [7]. From this new type of SLLN, we can obtain the Marcinkiewicz-Zygmund strong law in the blockwise M-dependence case. The following notation will be used throughout this paper. For x, let x denote the greatest integer less than or equal to x. For a, b R, min{a, b} max{a, b} will be denoted, respectively, by a b a b. We use Log to denote the logarithm to the base 2. The symbol C denotes a generic constant ( < C < ) which is not necessarily the same one in each appearance. The plan of the paper is as follows. Technical definitions, notation, the lemmas used in the proofs of the main results or their corollaries are consolidated into Section 2. The main results are stated proved in Section 3. In Section 4, some corollaries an interesting example are presented. 2. Preinaries Some definitions, notation, preinary results will be presented prior to establishing the main results. Although the concept of blockwise M-dependence was introduced by Móricz, Stadtmuller Thalmaier [6], we now recall this concept in a different way. Let M be a nonnegative integer. A finite collection of rom variables {X,...,X mn } is said to be M-dependent if either m n M + or m n > M + the rom variables {X,...,X ij } are independent of the rom variables {X,...,X mn } whenever (k i) (l j) > M, (k > i,l > j). A double array of rom variables {X mn,m,n } is said to be M-dependent if for each m,n, the rom variables {X,...,X mn } are M-dependent. Let {ω(k), k } {ν(k), k } be strictly increasing sequences of positive integers with ω() = ν() = set = [ ω(k),ω(k + ) ) [ ν(l),ν(l + ) ). We say that an array {X ij,i,j } of rom variables is blockwise M-dependent with respect to the blocks {,k,l }, if for each k l, the rom variables {X ij,(i,j) } are M-dependent. Thus 2

3 the rom elements with indices in each block are M-dependent but there are no independence requirements between the rom elements with indices in different blocks; even repetitions are permitted. For {ω(k),k }, {ν(k),k } {,k,l } as above, for m,n,k,l, we introduce the following notation: (mn) = {(i,j) : 2 m i < 2 m+,2 n j < 2 n+ }, (mn) = (mn), I mn = {(k,l) : (mn) }, r (m) k = min{m : m [ω(k),ω(k + )) [2 m,2 m+ )}, s (n) l = min{n : n [ν(l),ω(l + )) [2 n,2 n+ )}, c mn = cardi mn, ϕ(k,l) = c ij I (ij)(k,l), i= j= φ(k,l) = max i k,j l ϕ(i,j), where I (ij) denotes the indicator function of the set (ij),i,j. The following lemma establish the maximal inequality for double sums of blockwise M- dependent rom variables which is due to Móricz, Stadtmuller Thalmaier [6]. Lemma 2.. (Móricz, Stadtmuller Thalmaier [6]). Let {X ij, i m, j n} be a collection of mn rom variables let p 2. If the collection {X ij, i m, j n} is M-dependent EX ij = for all i m, j n, then E ( max S ) p m C k m, l n E X ij p, where S = k i= l j= X ij, k m, l n, the constant C is independent of m n. 25). The next lemma is a double sum analogue of the Toeplitz lemma (see, e.g., Loève [4], p. Lemma 2.2. Let {a mnij, i m, j n,m,n } be an array of positive constants such that sup m,n a mnij C < If {x mn,m,n } is a double array of constants a mnij = for every fixed i,j. x mn =, 3

4 then a mnij x ij =. Proof. For given ǫ >, there exists n such that x ij < ǫ C for all i j n. So that a mnij x ij a mnij x ij + ǫ. i j<n The conclusion of the lemma follows upon letting m n then ǫ. The following lemma establishes the strong law of large numbers for double arrays of arbitrary rom variables. Lemma 2.3. (Thanh [8]). Let {X ij,i,j } be a double array of rom variables let α >,β >. If then 3. Main results E X ij p (m α n β < for some < p, ) p m α n β X ij = a.s. Theorem 3.. Let {X ij,i,j } be a double array of mean rom variables let {a n,n } {b n,n } be nondecreasing sequences of positive constants such that a inf 2 n+ n a 2 n a sup 2 n+ n a 2 n b >, inf 2 n+ n b 2 n b <, sup 2 n+ n b 2 n >, (3.) <. (3.2) If {X ij,i,j } is blockwise M-dependent with respect to the blocks {,k,l }, if then Proof. Set T mn = T (mn) = max (p,q) (mn) E X ij p < for some p 2, (3.3) (a i b j ) p a m b n (φ(m,n)) (p )/p p i=r (m) k q j=s (n) l X ij = a.s. (3.4) X ij,(k,l) Imn,m,n (a 2 m+ a 2 m)(b 2 n+ b 2 n)(φ(2 m,2 n )) (p )/p 4 T (mn),m,n.

5 For m,n E(T mn ) p C (a 2 m+b 2 n+) p (φ(2 m,2 n )) p cp mn = = C C (a 2 m+b 2 n+) p C (a 2 m+b 2 n+) p E(T (mn) ) p C (a 2 m+b 2 n+) p 2 m+ i=2 m 2 n+ (i,j) (mn) (i,j) (mn) E X ij p E X ij p (a j=2 n i b j ) p. E(T (mn) ) p (by (3.) E X ij p (by Lemma 2.). It follows from (3.3) that i= j= E(T ij) p < so by the Markov inequality the Borel-Cantelli lemma T mn a.s. as m n. (3.5) Note that for m,n, letting k,l be such that 2 k m < 2 k+,2 l n < 2 l+, m n X ij a m b n (φ(m,n)) (p )/p k l (a 2 i+ a 2 i)(b 2 j+ b 2 j) T ij. (3.6) a 2 kb 2 l It follows from (3.2) that i= j= k l i= j= (a 2 i+ a 2 i)(b 2 j+ b 2 j) a 2 kb 2 l < C for all k,l. (3.7) The conclusion (3.4) follows from (3.5), (3.6), (3.7) Lemma 2.2. We now present an alternative version of Theorem 3.. When M =, a n m, b n n p = 2, Theorem 3.2 was obtained by Quang Thanh [7]. Theorem 3.2. Let {X ij,i,j } be a double array of mean rom variables let {a n,n } {b n,n } be nondecreasing sequences of positive constants such that (3.) (3.2) hold. If {X ij,i,j } is blockwise M-dependent with respect to the blocks {,k,l }, if then E X ij p (a i b j ) p (ϕ(i,j))p < for some p 2, (3.8) a m b n X ij = a.s. (3.9) 5

6 Proof. Define T (mn),(k,l) I m,m,n as in the proof of Theorem 3. set τ = (a 2 m+ a 2 m)(b 2 n+ b 2 n),m,n. For m,n E(τ mn ) p C C T (mn) (a 2 m+b 2 n+) pcp mn E(T (mn) C (a 2 m+b 2 n+) pcp mn 2 m+ i=2 m 2 n+ ) p (by (3.) (i,j) (mn) E X ij p (a j=2 n i b j ) p (ϕ(i,j))p. E X ij p (by Lemma 2.). It follows from (3.8) that i= j= E(τ ij) p <. The rest of the argument is exactly the same as that at the end of the proof of Theorem 3.. By using Theorem 3., we can now establish the Marcinkiewicz-Zygmund strong law of large numbers for double arrays of blockwise M-dependent rom variables. Theorem 3.3. Let {X ij,i,j } be a double array of rom variables which are stochastic dominated by a rom variable X in the sense that P { X ij > t} CP { X > t}, t,i,j. If {X ij,i,j } is blockwise M-dependent with respect to the blocks {,k,l }, if E( X r log + X ) < for some r < 2, (3.) then Proof. Set (mn) /r (φ(m,n)) /2 Y ij = X ij I( X ij (ij) /r ),i,j, Z ij = X ij I( X ij > (ij) /r ),i,j. (X ij EX ij ) = a.s. (3.) According to the proof of Lemma 2.2 of Gut [3], E(Y ij EY ij ) 2 (ij) 2/r <. (3.2) By using Theorem 3. with a n = b n = n /r,n p = 2, it follows from (3.2) that (mn) /r (φ(m,n)) /2 (Y ij EY ij ) = a.s. (3.3) 6

7 If r =, then P {X ij Y ij } = P { X ij > ij} C P { X > ij} < since E( X log + X ) <. So by the Borel - Cantelli lemma (X ij Y ij ) = a.s. (3.4) mn On the other h, mn E(X ij Y ij ) = mn It follows from (3.3), (3.4) (3.5) that mn(φ(m,n)) /2 mn C mn EZ ij E Z ij E( X I( X > ij)) as m n. (3.5) (X ij EX ij ) = a.s. If < r < 2, similarly (3.2), we also have E Z ij EZ ij (ij) /r <. (3.6) By using Lemma 2.3 with α = β = /r p =, it follows from (3.6) that (mn) /r (Z ij EZ ij ) = a.s. (3.7) The conclusion (3.) follows immediately from (3.3) (3.7). If m n in Theorem 3.3 take different powers, we get the following theorem. Theorem 3.4. Let q < r < 2 let {X ij,i,j } be a double array of rom variables which are stochastic dominated by a rom variable X. If {X ij,i,j } is blockwise M-dependent with respect to the blocks {,k,l }, if E( X r ) <, (3.8) then m /q n /r (φ(m,n)) /2 (X ij EX ij ) = a.s. (3.9) 7

8 Proof. Set Y ij = X ij I( X ij i /q j /r ),i,j, Z ij = X ij I( X ij > i /q j /r ),i,j. Firstly, m= n= E(Y mn EY mn ) 2 (m /q n /r ) 2 < = m= n= m= n= C = C = C m= n= m= n= = CE( X r ) EY 2 mn (m /q n /r ) 2 (m /q n /r ) 2 m /q n /r m /q n /r 2xP { X mn > x}dx (m /q n /r ) 2 2xP { X > x}dx P { X > t /2 m /q n /r }dt ( ( P { X t /2 m /q > n/r})) dt m= n= ( t r/2 m= ) m r/q dt <. (3.2) By using Theorem 3. with a n = n /q,b n = n /r,n p = 2, it follows from (3.2) that Next, m= n= E Z mn EZ mn m /q n /r 2 m /q n /r (φ(m,n)) /2 = 2 = C + C m= n= m= n= m= n= m= n= m= n= m= n= E Z mn m /q n /r m /q n /r m /q n /r m /q n /r m /q n /r m /q n /r (Y ij EY ij ) = a.s. (3.2) P { Z mn > x}dx m /q n /r P { X mn > m /q n /r }dx m /q n /r P { X mn > x}dx m /q n /r P { X > m /q n /r }dx m /q n /r P { X > x}dx 8

9 = C + C = C { X P m= n= m= n= E X r m r/q + C m= C + C > n/r} m/q ( P { X > tm /q n /r }dt m= C + CE X r( m= ( E X r ) m r/q t r dt m= n= { X P m /q t > n/r}) dt ) m r/q dt <. (3.22) tr By using Lemma 2.3 with α = /q,β = /r p =, it follows from (3.22) that m /q n /r The conclusion (3.9) follows immediately from (3.2) (3.23). (Z ij EZ ij ) = a.s. (3.23) 4. Corollaries example. In this section, we discuss some particular cases of Theorems An illustrative example is also provided. Part (i) Corollary 4. is the SLLN for a double array of blockwise M-dependent rom variables obtained by Móricz, Stadtmuller Thalmaier [6], Part (ii) of this corollary extends the Marcinkiewicz-Zygmund SLLN for double array to the case of blockwise M-dependence. Corollary 4.. Let {X ij,i,j } be a double array of rom variables which is blockwise M-dependent with respect to the blocks {[2 k,2 k+ ) [2 l,2 l+ ),k,l } (or, more generally, with respect to the blocks {[ω k,ω k+ ) [ν l,ν l+ ),k,l } where ω k = ν k = q k for all large k q > ), let {a n,n } {b n,n } be nondecreasing sequences of positive constants such that (3.) (3.2) hold. (i) If the rom variables {X ij,i,j } have mean (3.3) holds, then a m b n X ij = a.s. (4.) (ii) If the rom variables {X ij,i,j } are stochastic dominated by a rom variable X such that (3.) holds, then (mn) /r (X ij EX ij ) = a.s. (4.2) (iii) If q < r < 2 the rom variables {X ij,i,j } are stochastic dominated by a rom variable X such that (3.8) holds, then m /q n /r (X ij EX ij ) = a.s. (4.3) 9

10 Proof. If ω k = ν k = 2 k, k, then c mn =, φ(m,n) =,m, n, (4.4) if ω k = ν k = q k for all large k where q >, then c mn = O() φ(m,n) = O(),m, n. (4.5) So, the conclusion of Part (i) follows directly from Theorem 3., the conclusion of Part (ii) follows directly from Theorem 3.3, the conclusion of Part (iii) follows directly from Theorem 3.4. Note that every double array of rom variables {V ij,i,j } is blockwise -dependent with respect to the blocks {[k,k+) [l,l+),k,l }. The following corollary establishes the SLLN for arbitrary double arrays. Corollary 4.2. Let {X ij,i,j } be a double array of rom variables. If the rom variables {X ij,i,j } are stochastic dominated by a rom variable X such that (3.) holds, then (mn) /r+/2 (X ij EX ij ) = a.s. (4.6) Proof. Note that the array {X ij,i,j } is blockwise -dependent with respect to the blocks {[k,k + ) [l,l + ),k,l }. In this case we have φ(m,n) = O(mn). So, the conclusion (4.6) follows immediately from Theorem 3.3. Similarly, we also obtain some more particular cases of Theorems by noting that: If ω k = ν k = 2 kα for all large k where < α <, then c mn = O((mn) ( α)/α ) φ(m,n) = O((LogmLogn) ( α)/α ). (4.7) If ω k = ν k = k α, k where α >, then c mn = O(2 (m+n)/α ) φ(m,n) = O((mn) /α ). (4.8) We close by presenting an example. This example illustrative Theorem 3.3 ( Corollary 4.2). It shows that for any ǫ >, the it (mn) /r (φ(m,n)) /2 (mn) ǫ (X ij EX ij ) can be infinite a.s. It also shows that Corollary 4. can fail if the blockwise M-dependence hypothesis is not satisfied. Example 4.3. Let X be a non degenerate rom variable with EX =,E X r < for all r < 2, let X mn = X, m,n. Then {X mn,m,n } is blockwise -dependent with respect to the blocks {[k, k + ) [l, l + ), k, l } but is not blockwise

11 M-dependent with respect to the blocks {[2 k,2 k+ ) [2 l,2 l+ ),k,l } for all fixed M. By Corollary 4.2, we obtain (mn) /r+/2 X ij = a.s. For arbitrary ǫ >, we can choose 2/( + 2ǫ) < r < 2. Then (mn) /r+/2 ǫ X ij = (mn)(r+2rǫ 2)/(2r) X = a.s. We also see that (mn) /r X ij = (mn) /r X, so (4.2) fails. Acknowledgments The authors are grateful to the referee for his careful reading the manuscript. Part of research of Le Van Thanh was conducted during short visit to the Abdub Salam International Centre for Theoretical Physics (ICTP) in summer 28 under Federation Arrangement between ICTP Department of Mathematics, Vinh University. This author is grateful to ICTP Department of Mathematics, Vinh University for supporting this affiliate position. References [] Chow, Y.S. Teicher, H. (997). Probability Theory: Independence, Interchangeability, Martingales, Third Edition. Springer-Verlag, New York. [2] Gaposhkin, V.F. (994). On the strong law of large numbers for blockwise independent blockwise orthogonal rom variables. Teor. Veroyatnost. i Primenen. 39, (in Russian). English translation in Theory Probab. Appl. 39 (995), [3] Gut, A. (978). Marcinkiewicz laws convergence rates in the law of large numbers for rom variables with multidimensional indices. Ann. Probab. 6, [4] Loève, M. (977). Probability Theory, Vol. I, Fourth Edition. Springer-Verlag, New York. [5] Móricz, F. (987). Strong it theorems for blockwise m-dependent blockwise quasiorthogonal sequences of rom variables. Proc. Amer. Math. Soc., [6] Móricz, F., Stadtmuller, U. Thalmaier, M. (28). Strong laws for blockwise M- dependent rom fields. J. Theore. Probab. 2,

12 [7] Quang, N. V. Thanh, L. V. (25). On the strong laws of large numbers for twodimensional arrays of blockwise independent blockwise orthogonal rom variables. Probab. Math. Statist. 25, [8] Thanh, L. V. (27). On the strong law of large numbers for d-dimensional arrays of rom variables. Electron. Comm. Probab. 2,