Complete convergence in mean for double arrays of random variables with values in Banach spaces

Complete convergence in mean for double arrays of random variables with values in Banach spaces tài liệu, giáo án, bài g...

59 (2014) APPLICATIONS OF MATHEMATICS No 2, 177–190 COMPLETE CONVERGENCE IN MEAN FOR DOUBLE ARRAYS OF RANDOM VARIABLES WITH VALUES IN BANACH SPACES Ta Cong Son, Dang Hung Thang, Hanoi, Le Van Dung, Da Nang (Received July 22, 2012) Abstract The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables) In 2006, Rosalsky et al introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p) In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces Keywords: complete convergence in mean; double array of random variables with values in Banach space; martingale difference double array; strong law of large numbers; p-uniformly smooth space MSC 2010 : 60B11, 60B12, 60F15, 60F25 Introduction Let E be a real separable Banach space with norm · and {Xn , n 1} a sequence of random variables taking values in E (E-valued r.v.’s for short) Recall that Xn is said to converge completely to in mean of order p if ∞ E Xn p < ∞ n=1 This mode of convergence was investigated for the first time by Chow [2] for the sequence of real-valued random variables and by Rosalsky et al [6] for the sequence The research of the first author (grant no 101.03-2013.02), second author (grant no 101.03-2013.02) and third author (grant no 10103-2012.17) have been partially supported by Vietnams National Foundation for Science and Technology Development (NAFOSTED) The research of the first author has been partially supported by project TN-13-01 177 of random variables taking values in a Banach space In this paper, we introduce and study the complete convergence in mean of order p to of double arrays of E-random variables In Section some properties of the complete convergence in mean of order p are given and a new characterization of a p-uniformly smooth Banach space E in terms of the complete convergence in mean of order p of double arrays of E-valued r.v.’s is obtained These results are used in Section to obtain some strong laws of large numbers for martingale difference double arrays of random variables taking values in Banach spaces Preliminaries and some useful lemmas For a, b ∈ R, max {a, b} will be denoted by a ∨ b Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same in each appearance The set of all non-negative integers will be denoted by N and the set of all positive integers by N∗ For (k, l) and (m, n) ∈ N2 , the notation (k, l) (m, n) (or (m, n) (k, l)) means that k m and l n Definition 2.1 Let E be a real separable Banach space with norm {Smn ; (m, n) (1, 1)} be an array of E-valued r.v.’s · and let c (1) Smn is said to converge completely to and we write Smn → if ∞ ∞ P ( Smn > ε) < ∞ for all ε > m=1 n=1 (2) Smn is said to converge to in mean of order p (or in Lp for short) as m∨n → ∞ Lp and we write Smn −→ as m ∨ n → ∞ if E Smn p → as m ∨ n → ∞ c,Lp Smn is said to converge completely to in mean of order p and we write Smn −→ if ∞ ∞ E Smn p < ∞ m=1 n=1 (3) Smn is said to converge almost surely to as m ∨ n → ∞ and we write Smn → a.s as m ∨ n → ∞ if P 178 lim m∨n→∞ Smn = = c,Lp Lp It is clear that Smn −→ implies Smn −→ as m ∨ n → ∞ By the Markov inequality ∞ ∞ P { Smn > ε} < ∞ for all ε > m=1 n=1 c,Lp c a.s we also see that Smn −→ implies Smn → and Smn −→ For an E-valued r.v X and sub σ-algebra G of F , the conditional expectation E(X | G) is defined and enjoys the usual properties (see [7]) A real separable Banach space E is said to be p-uniformly smooth (1 p 2) p if there exists a finite positive constant C such that for any L integrable E-valued martingale difference sequence {Xn , n 1}, n E n p Xi C i=1 E Xi p i=1 Clearly every real separable Banach space is 1-uniformly smooth and every Hilbert space is 2-uniformly smooth If a real separable Banach space is p-uniformly smooth for some < p then it is r-uniformly smooth for all r ∈ [1, p) For more details, the reader may refer to Pisier [5] Let {Xmn , (m, n) (1, 1)} be a double array of E-valued r.v.’s, let Fij be the σ-field generated by the family of E-random variables {Xkl ; k < i or l < j} and F11 = {∅ ; Ω} The array of E-valued r.v.’s {Xmn , (m, n) (1, 1)} is said to be an E-valued martingale difference double array if E(Xmn | Fmn ) = for all (m, n) (1, 1) The following lemmas are necessary for proving the main results in the paper Lemma 2.1 Let E be a p-uniformly smooth Banach space for some p and let {Xmn ; (m, n) (1, 1)} be a double array of E-valued r.v.’s satisfying E(Xij | Fij ) which is measurable with respect to Fmn for all (i, j) (m, n) Then k l k l m n m p (Xij − E(Xij | Fij )) E max i=1 j=1 n C E Xij p , i=1 j=1 where the constant C is independent of m and n P r o o f The proof is completely similar to that of Lemma of Dung et al [3] k l after replacing Skl = k Vij by Skl = i=1 j=1 l (Xij − E(Xij | Fij )) i=1 j=1 The following lemma is a version of Lemma of Adler and Rosalsky [1] for arrays of positive constants 179 Lemma 2.2 Let p > and let {bmn ; (m, n) (1, 1)} be an array of positive p p constants with bij /ij bmn /mn for all (i, j) (m, n) and lim bpmn /mn = ∞ m∨n→∞ Then ∞ ∞ mn as m ∨ n → ∞ p = O p b b mn i=m j=n ij if and only if bprm,sn > rs p m∨n→∞ bmn lim inf for some integers r, s bp mn P r o o f Set cmn = mn , (m, n) (1, 1) then cij lim cmn = ∞ It is required to show that cmn for all (i, j) (m, n) and m∨n→∞ ∞ ∞ (2.1) i=m j=n 1 =O ijcij cmn as m ∨ n → ∞ if and only if (2.2) lim inf m∨n→∞ crm,sn >1 cmn for some integers r, s If (2.2) holds, then exits δ > and no ∈ N such that crm,sn so ∞ ∞ i=m j=n ijcij ∞ mr k+1 −1 nsl+1 −1 k,l=0 i=mr k (r − 1)(s − 1) j=nsl ∞ cmn ∞ klckl k=1 δk k,l=0 δcmn for all m∨n no , (r − 1)(s − 1) cmrk ,nsl Then, we have (2.1) Conversely, assume that (2.2) does not hold Then lim inf crm,sn /cmn = for m∨n→∞ any r, s 2, then crm,sn < 2cmn for any r, s and an infinite numbers pair of values of (m, n) and so, ∞ ∞ i=m j=n mr ns 1 > ijcij ijc ij i=m j=n (log r)(log s) (log r)(log s) > crm,sn 2cm,n Since r, s is arbitrary, (2.1) does not hold as well 180 The complete convergence in mean From now on, E be a real separable Banach space and for each double array of E-valued r.v.’s {Xmn ; (m, n) (1, 1)}; we always denote Fij is σ-field generated by the family of E-random variables {Xkl ; k < i or l < j}, F11 = {∅ ; Ω}, k l k l ∗ Xij and Skl = Skl = i=1 j=1 (Xij − E(Xij | Fij )); i=1 j=1 {bmn ; (m, n) (1, 1)} be a sequence of positive constants satisfying bij bmn for all (i, j) (m, n) and lim bmn = ∞ m∨n→∞ Firstly, we show a condition under which the complete convergence in mean order p implies the convergence a.s and the convergence in Lp Theorem 3.1 Let {Xmn ; (m, n) Suppose that (3.1) (1, 1)} be a double array of E-valued r.v.’s M = sup m,n b2m+1 2n+1 < ∞ b2m 2n If (3.2) max(k,l) (m,n) Skl c,Lp −→ (mn)1/p bmn for some p 2, then (3.3) max(k,l) (m,n) Skl bmn → a.s and in Lp as m ∨ n → ∞ P r o o f Set Amn = {(k, l), (2n , 2m ) (3.4) E max(k,l) (2m ,2n ) (k, l) ≺ (2m+1 , 2n+1 )} We see that Skl p b2m 2n (m,n) (0,0) E (m,n) (0,0) Mp M max(k,l) (2m ,2n ) Skl b2m+1 2n+1 (m,n) (0,0) (k.l)∈Amn E max(i,j) (m,n) (0,0) (k,l)∈Amn (k,l) Sij p bkl Mp p 2m 2n E max(i,j) (k,l) Sij p bkl 181 Mp (m,n) (0,0) (k,l)∈Amn 4M p (m,n) (1,1) 4M p max(i,j) (k,l) Sij E kl bkl max(k,l) (m,n) Skl E mn bpmn max(k,l) (m,n) Skl (mn)1/p bmn E (m,n) (1,1) p p p < ∞ This implies that (3.5) E max(k,l) (2m ,2n ) Skl p b2m 2n →0 as m ∨ n → ∞ Now for (k, l) ∈ Anm we have (3.6) E max(i,j) E Sij (k,l) bkl max(k,l) p E (2m+1 ,2n+1 ) max(k,l) (2m+1 ,2n+1 ) Skl p bkl Skl p M pE b2m 2n max(k,l) Skl b2m+1 2n+1 k From (3.5) and (3.6) we conclude that (2m+1 ,2n+1 ) l sup Xij p Lp /bmn −→ as (k,l) (m,n) j=1 i=1 m ∨ n → ∞ By (3.4) and the Markov inequality, for all ε > we have P (m,n) (0,0) max Skl (k,l) (2m ,2n ) 4M p εp E (m,n) (1,1) εb2m 2n max(k,l) (m,n) Skl (mn)1/p bmn p < ∞ This implies by the Borel-Cantelli lemma that max(k,l) (2m ,2n ) Skl b2m 2n a.s −→ as m ∨ n → ∞ By the same argument as in (3.6), we have sup(k,l) (m,n) k j=1 l i=1 bmn Xij a.s −→ as m ∨ n → ∞ The proof of the theorem is completed The following theorem shows that the rate of the convergence of strong laws of large numbers may be obtained as a consequence of the complete convergence in mean 182 Theorem 3.2 Let α, β ∈ R and let {Xmn ; (m, n) E-valued r.v.’s If (mα nβ )1/p bmn c,Lp Skl −→ max (k,l) (m,n) (1, 1)} be a double array of for some p 2, then m−α n−β P b−1 mn (3.7) (m,n) (1,1) max (k,l) (m,n) Skl > ε < ∞ for every ε > In the case of α < 1, β < and {bmn ; (m, n) P sup (k,l) (m,n) (1, 1)} satisfying (3.1), (3.7) implies Skl >ε =o 1−α bkl m n1−β as m ∨ n → ∞ for every ε > P r o o f By Markov inequality, for all ε > m−α n−β P b−1 mn (m,n) (1,1) εp max (k,l) (m,n) Skl ε m−α n−β E max(k,l) (m,n) Skl p bmn (m,n) (1,1) < ∞ Then, we have (3.7) Let α < 1, β < Fix ε > 0, and set Amn = {(k, l), (2n−1 , 2m−1 ) ≺ (k, l) m n (2 , )} We see that m−α n−β P sup (k,l) (m,n) (m,n) (1,1) i −1 b−1 kl Skl > ε 2j −1 m−α n−β P = sup (k,l) (m,n) (i,j) (1,1) m=2i−1 n=2j−1 2i −1 2j −1 m=2i−1 n=2j−1 2−iα 2−jβ P C (i,j) (1,1) (i,j) (1,1) sup (i,j) (1,1) max (u,v) (i,j) (k,l)∈Auv 2i(1−α) 2j(1−β) C sup (k,l) (2i−1 ,2j−1 ) 2i(1−α) 2j(1−β) P C b−1 kl Skl > ε b−1 kl Skl > ε P b−1 2u−1 2v−1 (u,v) (i,j) b−1 kl Skl > ε max (k,l) (2u ,2v ) Skl > ε 183 P b−1 2u−1 2v−1 C max (u,v) (1,1) C u(1−α) v(1−β) P b−1 2u 2v (u,v) (1,1) m−α n−β P b−1 mn C (m,n) (1,1) Since P sup (k,l) (m,n) sup (k,l) (m,n) (i,j) (u,v) maxu (k,l) (2 max ,2v ) Skl > (k,l) (m,n) ∗2 b−1 kl Skl > ε , (m, n) ∈ N Skl > ε M ε M < ∞ (by (3.7)) are non-increasing in (m, n) for in N∗2 , it follows that order relationship P 2i(1−α) 2j(1−β) Skl > ε (k,l) (2u ,2v ) b−1 kl Skl > ε = o m1−α n1−β as m ∨ n → ∞ for all ε > Now we establish sufficient conditions for complete convergence in mean of order p Theorem 3.3 Let E be a p-uniformly smooth Banach space for some p Let {Xmn ; (m, n) (1, 1)} be a double array of E-valued r.v.’s such that E(Xij |Fij ) is measurable with respect to Fmn for all (i, j) (m, n) Suppose that ∞ ∞ b−p mn < ∞ (3.8) m=1 n=1 If ∞ ∞ (3.9) ϕ(m, n)E Xmn p < ∞, m=1 n=1 ∞ ∞ where ϕ(m, n) = i=m j=n b−p ij , then (3.10) bmn (k,l) (m,n) p ∞ c,Lp ∗ Skl −→ max P r o o f We have ∞ ∞ E m=1 n=1 max(k,l) (m,n) bpmn ∗ Skl C n j=1 E bpmn m=1 n=1 ∞ ∞ C E Xij i=1 j=1 ∞ ∞ C ∞ Xij p ∞ p m=i n=j ϕ(i, j)E Xij i=1 j=1 184 m i=1 ∞ p (by Lemma 2.1) bpmn < ∞ (by (3.9)) A characterization of p-uniformly smooth Banach spaces in terms of the complete convergence in mean of order p is presented in the following theorem Theorem 3.4 Let p 2, let E be a real separable Banach space Then the following statements are equivalent: (i) E is of p-uniformly smooth (ii) For every double array of random variables {Xmn ; (m, n) (1, 1)} with values in E such that E(Xij | Fij ) is measurable with respect to Fmn for all (i, j) (m, n), and every double array of positive constants {bmn ; (m, n) (1, 1)} with bij bmn for all (i, j) (m, n) and satisfying ∞ ∞ mn , p =O p b b mn i=m j=n ij (3.11) the condition ∞ ∞ (3.12) mn m=1 n=1 E Xmn bpmn p 1, hence E(Xmn | Fmn ) = for all (m, n) (1, 1), we have n c,Lp i=1 Xi −→ (mn)(p+1)/p 0, n a.s Xi /mn −→ as m ∨ n → ∞ and by Theorem 3.1 (with bmn = mn) then i=1 n Taking m = and letting n → ∞, we obtain that 1/n Xi → a.s i=1 Then by Theorem 2.2 in [4], E is p-uniformly smooth For bmn = mα+1/p nβ+1/p (α, β > 0), from (ii) of Theorem 3.4 we get the following corollary Corollary 3.1 Let E be a p-uniformly smooth Banach space for some p Let α, β > and let {Xmn ; (m, n) (1, 1)} be an array of E-valued r.v.’s such that E(Xij | Fij ) is measurable with respect to Fmn for all (i, j) (m, n) If ∞ ∞ E Xmn p < ∞, nαp mβp m=1 n=1 then 186 ∗ (m,n) Skl mα+1/p nβ+1/p sup(k,l) c,Lp −→ 4 Applications to the strong law of large numbers By applying the theorems about complete convergence in mean in Section we establish some results on strong laws of large numbers for double arrays of martingale differences with values in p-uniformly smooth Banach spaces Theorem 4.1 Let E be a p-uniformly smooth Banach space for some p and let {Xmn , (m, n) (1, 1)} be an E-valued martingale differences double array If ∞ ∞ E Xmn p < ∞, nαp mβp m=1 n=1 then max(k,l) (m,n) Skl → a.s and in Lp as m ∨ n → ∞ mα n β P r o o f By Corollary 3.1, we have sup(k,l) (m,n) Skl mα+1/p nβ+1/p c,Lp −→ Applying Theorem 3.1 with bmn = mα nβ , we have max(k,l) (m,n) Skl → a.s and in Lp as m ∨ n → ∞ mα n β The following theorem is a Marcinkiewicz-Zygmund type law of large numbers for double arrays of martingale differences Theorem 4.2 Let r s < q < p 2, let E be a p-uniformly smooth Banach space Suppose that {Xmn , (m, n) (1, 1)} is an E-valued martingale differences double array which is stochastically dominated by an E-random variable X in the sense that for some < C < ∞, P { Xmn x} CP { X x} for all (m, n) (1, 1) and x > If E(Xij I( Xij i1/q j 1/r ) | Fij ) is measurable with respect to Fmn for all (i, j) (m, n) and E X q < ∞ then (4.1) max(k,l) (m,n) Skl → a.s and in Ls as m ∨ n → ∞ m1/q n1/r 187 P r o o f For each (m, n) (1, 1) set m1/q n1/r ), Zmn = Xmn I( Xmn > m1/q n1/r ), Ymn = Xmn I( Xmn Umn = Ymn − E(Ymn | Fmn ), Vmn = Zmn − E(Zmn | Fmn ) It is clear that Xmn = Umn + Vmn First, ∞ ∞ E Ymn p (m1/q n1/r )p m=1 n=1 ∞ m1/q n1/r ∞ 1/q (m n1/r )p m=1 n=1 ∞ C 1/q n1/r )p (m m=1 n=1 ∞ m1/q n1/r ∞ ∞ =C P { X > t1/p m1/q n1/r } dt ∞ P =C n=1 = CE( X q pxp−1 P { X > x} dx 0 m=1 n=1 ∞ pxp−1 P { Xmn > x} dx m=1 ) tq/p X t1/p n1/r > m1/q dt ∞ q/r n n=1 dt < ∞ By applying Corollary 3.1, it follows that k l (m,n) i=1 j=1 m1/q+1/p n1/r+1/p sup(k,l) Uij c,Lp −→ 0, and by Theorem 3.1, we get sup(k,l) k (m,n) i=1 m1/q n1/r l j=1 Uij → a.s and in Lp as m ∨ n → ∞ Then sup(k,l) (4.2) k (m,n) i=1 m1/q n1/r l j=1 Uij → a.s and in Ls as m ∨ n → ∞ Next, ∞ ∞ ∞ ∞ E Zmn s = 1/q n1/r )s s/q ns/r (m m m=1 n=1 m=1 n=1 188 ∞ sxs−1 P { Zmn > x} dx ∞ = m1/q n1/r ∞ s/q m ns/r m=1 n=1 ∞ ∞ ∞ + m=1 n=1 ∞ ∞ sxs−1 P { Xmn > x} dx ms/q ns/r m1/q n1/r m1/q n1/r C s/q m ns/r m=1 n=1 ∞ s/q ns/r m=1 m n=1 ∞ ∞ =C xs−1 P { X > m1/q n1/r } dx ∞ ∞ +C sxs−1 P { Xmn > m1/q n1/r } dx xs−1 P { X > x} dx m1/q n1/r X > m1/q n1/r P n=1 m=1 ∞ ∞ ∞ ts−1 P { X > tm1/q n1/r } dt + m=1 n=1 ∞ C E X nq/r n=1 ∞ C E X nq/r n=1 ∞ CE X q ∞ ∞ q ts−1 + ∞ 1 X P n1/r t n=1 m=1 ∞ ∞ q nq/r n=1 ∞ ts−1 + E X q nq/r tq n=1 dt + tq−s+1 > m1/q dt dt < ∞ By applying Corollary 3.1, it follows that k l (m,n) i=1 j=1 m1/q+1/s n1/r+1/s sup(k,l) Vij c,Ls −→ and by Theorem 3.1 we have (4.3) sup(k,l) k (m,n) i=1 m1/q n1/r l j=1 Vij → a.s and in Ls as m ∨ n → ∞ By (4.2), (4.3) and since the inequality E X + Y s 2s−1 (E X for s we have (4.1) The proof is completed s +E Y s ) holds Finally, we establish the rate of convergence in the strong law of large numbers Theorem 4.3 Let < r < p, < s < p, let E be a p-uniformly smooth Banach space for some p and {Xmn ; (m, n) (1, 1)} an E-valued martingale differences double array If ∞ ∞ E Xmn p < ∞, np−r mp−s m=1 n=1 189 then (4.4) P sup (k,l) (m,n) Skl >ε =o r kl m ns as m ∨ n → ∞ for every ε > P r o o f By (ii) in Theorem 3.4 and Lemma 2.2 (with {bmn = m1+(1−r)/p × n1+(1−s)/p ; (m, n) (1, 1)}), we have max m1+(1−r)/p n1+(1−s)/p (k,l) (m,n) c,Lp Skl −→ 0, and by Theorem 3.2 (with α = − r, β = − s), we have (4.4) References [1] A Adler, A Rosalsky: Some general strong laws for weighted sums of stochastically dominated random variables Stochastic Anal Appl (1987), 1–16 [2] Y S Chow: On the rate of moment convergence of sample sums and extremes Bull Inst Math., Acad Sin 16 (1988), 177–201 [3] L V Dung, T Ngamkham, N D Tien, A I Volodin: Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces Lobachevskii J Math 30 (2009), 337–346 [4] J Hoffmann-Jørgensen, G Pisier: The law of large numbers and the central limit theorem in Banach spaces Ann Probab (1976), 587–599 [5] G Pisier: Martingales with values in uniformly convex spaces Isr J Math 20 (1975), 326–350 [6] A Rosalsky, L V Thanh, A I Volodin: On complete convergence in mean of normed sums of independent random elements in Banach spaces Stochastic Anal Appl 24 (2006), 23–35 [7] F S Scalora: Abstract martingale convergence theorems Pac J Math 11 (1961), 347–374 Authors’ addresses: Ta Cong Son, Dang Hung Thang, Hanoi University of Science, Hanoi, Vietnam, e-mails: congson82@gmail.com, hungthang.dang@gmail.com; Le Van Dung, Da Nang University of Education, Da Nang, Vietnam, e-mail: lvdunght@gmail.com 190 zbl MR zbl MR zbl MR zbl MR zbl MR zbl MR zbl MR .. .of random variables taking values in a Banach space In this paper, we introduce and study the complete convergence in mean of order p to of double arrays of E -random variables In Section... properties of the complete convergence in mean of order p are given and a new characterization of a p-uniformly smooth Banach space E in terms of the complete convergence in mean of order p of double arrays. .. arrays of E-valued r.v.’s is obtained These results are used in Section to obtain some strong laws of large numbers for martingale difference double arrays of random variables taking values in Banach