Statistics and Probability Letters 83 (2013) 1901–1910 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro The Brunk–Prokhorov strong law of large numbers for fields of martingale differences taking values in a Banach space Ta Cong Son ∗ , Dang Hung Thang Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam article abstract info Article history: Received 19 November 2012 Received in revised form 16 April 2013 Accepted 21 April 2013 Available online May 2013 In this paper, we define a new type of fields of martingale differences taking values in Banach spaces and establish the Brunk–Prokhorov strong laws of large numbers and the convergence rate in the strong laws of large numbers for such fields © 2013 Elsevier B.V All rights reserved MSC: 60B11 60B12 60F15 60F25 60G42 Keywords: p-uniformly smooth Banach spaces Field of martingale differences Strong law of large numbers Introduction and preliminaries Let q ≥ and {Xn ; n ≥ 1} be a sequence of independent random variables The Brunk–Prokhorov strong law of large (Brunk–Prokhorov SLLN) (see Brunk, 1948; Prokhorov, 1950) stated that if EXn = 0, for all n ≥ and ∞ numbers 2q q +1 E | X | / n < ∞, then n n =1 lim n→∞ n 1 n k =1 Xk = a.s Brunk–Prokhorov SLLN was extended to martingale differences e.g in Fazekas and Klesov (2000) and Hu et al (2008) For the field of random variables with multidimensional index, Lagodowski (2009) established the Brunk–Prokhorov SLLN for fields of independent E-valued random variables and Noszaly and Tomacs (2000) proved the Brunk–Prokhorov SLLN for fields of real-valued martingale differences In this paper, we introduce a new type of fields of E-valued martingale differences and establish the Brunk–Prokhorov SLLN for such fields In Section 1, a new type of fields of E-valued martingale differences is defined, illustrated by some non-trivial examples and compared with the usual definition In Section we prove some useful lemmas and inequalities Section contains the main results including the Brunk–Prokhorov SLLN for such fields of E-valued martingale differences Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance ∗ Corresponding author Tel.: +84 1989318669 E-mail addresses: congson82@gmail.com, congson82@hus.edu.com.vn (T.C Son), hungthang.dang53@gmail.com (D.H Thang) 0167-7152/$ – see front matter © 2013 Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.spl.2013.04.023 1902 T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 Let E be 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 E-valued martingales {Sn ; ≤ n ≤ m} E ∥Sm ∥p ≤ C m E ∥Sn − Sn−1 ∥p (1.1) n =1 Clearly every real separable Banach space is of 1-uniformly smooth, the real line (the same as any Hilbert space) is of 2-uniformly smooth and the space Lp (1 ≤ p ≤ 2) is of p-uniformly smooth If a real separable Banach space of p-uniformly smooth (1 < p ≤ 2) then it is of r-uniformly smooth for all r ∈ [1, p) Using classical methods from martingale theory, it was shown that (see Woyczyn’ski, 1978) if E is of p-uniformly smooth, then for all ≤ q < ∞ there exists a finite constant C such that q E ∥Sm ∥ ≤ CE pq m ∥Si − Si−1 ∥ p (1.2) i =1 Let d be a positive integer For m = (m1 , , md ), n = (n1 , , nd ) ∈ Nd , denote m + n = (m1 + n1 , , md + nd ), m − n = (m1 − n1 , , md − nd ), |n| = n1 n2 nd , ∥n∥ = min{n1 , , nd }, = (1, , 1) ∈ Nd , di=1 (mi < ni ) means that there is at least one of m1 < n1 , m2 < n2 , , md < nd holds We write m ≼ n (or n ≽ m) if mi ≤ ni , ≤ i ≤ d; m ≺ n if m ≼ n d and m ̸= n; m ≪ n (or n ≫ m) if i=1 (mi < ni ) Let (Ω , F , P ) be a probability space, E be a read separable Banach space, and B (E) be the σ -algebra of all Borel sets in E Definition Let {Xn , ≼ n ≼ N} be a field of E-valued random variables and {Fn , ≼ n ≼ N} be a field of nondecreasing sub-σ -algebras of F with respect to the partial order ≼ on Nd The field {Xn , Fn , ≼ n ≼ N} is said to be an adapted field if Xn is Fn -measurable for all ≼ n ≼ N The adapted field {Xn , Fn , ≼ n ≼ N} is said to be a field of martingale differences in the usual sense if E (E (X |Fm )|Fn ) = E (X |Fm∧n ) for all X ∈ L1 (1) E (Xn |Fn ) = (2) and for all ≼ n ≼ N (See Christofides and Serfling, 1990; Lagodowski, 2009) The adapted field {Xn , Fn , ≼ n ≼ N} is said to be a field of martingale differences if E (Xn |Fn∗ ) = for all ≼ n ≼ N where Fn = σ {Fl : ∨ ∗ d i =1 (3) (li < ni )}, for ≼ n ≼ N (see Son et al., 2012) Remark For a field of martingale differences the condition (1) about {Fn , ≼ n ≼ N} is not required but the condition (3) seems to be stronger than the condition (2) Example Let {Xn , ≼ n ≼ N} be a field of independent random variables with mean Put Fn = σ (Xk , k ≼ n), then E (Xn |Fn∗ ) = and ≼ n ≼ N Therefore, {Xn , Fn , ≼ n ≼ N} is a field of martingale differences Example Let {Xn , Gn : n ≥ 1} is a sequences of martingale differences, set Xn = Xn if n = (n, n, , n) and Xn = if n ̸= (n, n, , n); Gn = Gn if n = (n, n, , n) and Gn = {∅, Ω } if n ̸= (n, n, , n) Let Fn = σ {Gk , k ≼ n} for all n ≽ 1, then {Xn , Fn : n ≽ 1} is a field of martingale differences, but it is not a field of independent random variables Example Let {Xn , ≼ n ≼ N} be a field of independent random variables with mean Put Fn = σ (Xk , k ≼ n) and Yn = k≼n Xk , if EYn < ∞ for all n ≼ N, then E (Yn |Fn∗ ) = and ≼ n ≼ N Therefore, {Xn , Fn , ≼ n ≼ N} is a field of martingale differences, but it is not a field of independent random variables At the same time, it is not a field of martingale differences in the usual sense since the condition (1) does not hold for {Fn , ≼ n ≼ N} Definition Let {an , n ∈ Nd } be a field of elements in E We say that an → a as n → ∞ if for any ϵ > there exists nϵ ∈ Nd such that for all n ≽ nϵ then ∥an − a∥ < ϵ We say that an → a strongly as n → ∞ if for any ϵ > there exists nϵ ∈ Nd such that for all n ̸≼ nϵ then ∥an − a∥ < ϵ (See Lagodowski, 2009) Clearly, an → a strongly as n → ∞ then an → a as n → ∞, but the converse is not true For example, let a ̸= b, a(n1 ,1 ,1) = b and an = a if otherwise, then an → a but an ̸→ strongly as n → ∞ It is easy to see that in the case d = 1, the strong convergence and the convergence are equivalent T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 1903 Some useful lemmas and inequalities Lemma 2.1 Let ≤ p ≤ 2, q ≥ and E be a real separable p-uniformly smooth Banach space Then there exists positive C such that for each field of E-valued martingale differences {Xk , Fk ; ≼ k ≼ N}, we have q q/p p E X ≤ CE ∥Xk ∥ 1≼k≼N k 1≼k≼N (2.1) Proof For the field of E-valued martingale differences {Xk , Fk ; ≼ k ≼ N}, we define a sequence {Yk , Gk ; ≤ k ≤ |N|} by putting Y1 = X(1, ,1,1) ; Y2 = X(1, ,1,2) ; ; YNd = X(1, ,1,Nd ) ; YNd +1 = X(1, ,2,1) ; YNd +1 = X(1, ,2,2) ; ; Y|N| = XN , and for all ≤ k ≤ |N|, let Gk = σ {Yi ; ≤ i ≤ k} Since E (Yk |Gk−1 ) = E (E (Xi |Fi∗ )|Gk−1 ) = 0, (where Yk = Xi ) so {Yk ; Gk ; ≤ k ≤ |N|} is a sequence of E-valued martingale differences From the inequality (1.2) we get (2.1) Lemma 2.2 Let q ≥ and E be a real separable Banach space For each field of E-valued martingale differences {Xk , Fk ; ≼ (s) k ≼ N}, put Fn = σ {Fk : k = (k1 , , kd ), ≤ ki ≤ Ni (i ̸= s), and ks = ns } for all ≤ s ≤ d and Sn = k≼n Xk Then we have E (∥S(n,s,ns +1) ∥q |Fn(s) ) ≥ ∥Sn ∥q (2.2) where S(n,s,α) = S(n1 , ,ns−1 ,α,ns+1 , ,nd ) (s) ⊂ F(∗i,s,ns +1) for all (i, s, 1) ≼ n = (n1 , , ns , , nd ), then (s) (s) ∗ E (S(n,s,ns +1) − Sn |Fn ) = E E (X(i,s,ns +1) |F(i,s,ns +1) )Fn = 0, Proof We have S(n,s,ns +1) − Sn = (i,s,1)≼n X(i,s,ns +1) , and Fn (i,s,1)≼n so E (∥S(n,s,ns +1) ∥q |Fn(s) ) ≥ ∥E (S(n,s,ns +1) |Fn(s) )∥q = ∥Sn ∥q Lemma 2.3 Let {Sn ; n ≽ 1} be a field of E-valued random variables Then, Sn → strongly a.s as |n| → ∞ if only if for all ε > 0, sup ∥Sk ∥ > ε lim P ∥n∥→∞ = (2.3) k̸≼n Sn → a.s as |n| → ∞ if only if for all ε > 0, lim P ∥n∥→∞ sup ∥Sk ∥ > ε = (2.4) k≽ n Proof Necessary Suppose that Sn → a.s as |n| → ∞ For each ϵ > 0, let Aϵn = supk̸≼n ∥Sk ∥ > ϵ and Aϵ = then P (Aϵ ) = 0, n = (n1 , n2 , , nd ) = (n1 , n1 ), by the continuity from below theorem, so ϵ = P (A ) = P ϵ An n ≽1 = lim P n1 →∞ ϵ An n ≽1 = · · · = lim P (Aϵn ), ∥n∥→∞ we have (2.3) Sufficient Suppose (2.3) holds, for all n ∈ Nd and i ∈ N, we let Ain = sup ∥Sk ∥ ≥ k̸≼n i and A = i≥1 n≽1 Ain n ≽1 Aϵn , 1904 T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 by the continuity from below theorem, then P (A) = P = lim lim P (Ain ) = lim = 0, Ain i→∞ ∥n∥→∞ i≥1 n≽1 i→∞ i.e Sn → a.s as |n| → ∞ The prove of (2) is similar to the one of (1) Theorem 2.4 Let E be a Banach space, {Xn , Fn ; ≼ n ≼ N} be a field of E-valued martingale differences and {bn , ≼ n ≼ N} be a field of positive constants such that bn ≤ bm for all n ≼ m Then for all ε > 0, q ≥ we have εP ∥Sn ∥q max ≥ϵ q bn 1≼n ≼N 1≼n≼N−1 bn ≤ q 1≤s≤d − E ∥Sn ∥q N−1≺n≼N bn q E ∥Sn ∥ + q b(n,s,ns +1) q where b(n,s,α) = b(n1 , ,ns−1 ,α,ns+1 , ,nd ) Proof Let ‘‘ 0, q ≥ 1, we have ε P sup n ≽1 ∥Sn ∥q q bn ≥ϵ ≤ 1≤s≤d where b(n,s,α) = b(n1 , ,ns−1 ,α,ns+1 , ,nd ) n ≽1 bn q − q b(n,s,ns +1) E ∥Sn ∥ q T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 1905 Proof By the condition (2.5), there exists field Nn ∈ Nd such that Nn ≺ Nn+1 for all n ∈ N, ∥Nn ∥ → ∞ and limn→∞ maxNn −1≺k≼Nn E ∥Sk ∥q = q bk By the continuity from below theorem and Theorem 2.4, we have ε P sup ∥Sn ∥q ≥ε q bn n ≽1 = lim ε P ∥S n ∥q sup n→∞ 1≼n≼Nn −1 bn ≤ lim q n→∞ 1≤s≤d = Remark If lim∥n∥→∞ E ∥Sn ∥q q bn − q 1≤s≤d ≥ε q bn 1≼n≼Nn bn n ≽1 − q E ∥S n ∥ + q b(n,s,ns +1) E ∥S n ∥q Nn −1≺n≼Nn bn q E ∥S n ∥ q b(n,s,ns +1) q = then (2.5) holds Corollary 2.6 Let ≤ p ≤ 2, q ≥ and E be a real separable p-uniformly smooth Banach space Then there exists positive C such that for all field of E-martingale difference {Xk ; ≼ k ≼ N}, we have εP max ∥Sn ∥ ≥ ε q q/p ≤ 1≼n≼N q E ∥Sn ∥ ≤ CE N−1≺n≼N ∥X n ∥ p 1≼n≼N Proof By Theorem 2.4 with bn = for all n ≽ and Lemma 2.1, we have εP max ∥Sn ∥q ≥ ε ≤ 1≼n≼N E ∥Sn ∥q N−1≺n≼N q/p ≤C E N−1≺n≼N ∥X k ∥ p ≼ k≼ n q/p d ≤ ·C ·E p ∥X n ∥ 1≼n ≼N Main results Theorem 3.1 Let q ≥ 1, E be a Banach space, {bn , n ≽ 1} and {cn , n ≽ 1} be a field positive constants such that bn ≤ bm for all n ≼ m, ξ (n), n ≽ be a positive function, {Xn , Fn ; n ≽ 1} be a field of E-valued martingale differences satisfying (2.5) and the following condition E ∥Sn ∥q ≤ C ξ (n) ck (3.1) 0, 1906 T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 where N = (N1 , , Nk , , Nd ) Put N(k) = (1, , Nk , , 1) By Corollary 2.5 and the condition (3.1) we have ∥Sn ∥ ϵ P sup q bn n̸≼N ≥ϵ =ϵ P q d sup ≤ϵ q d P sup ≤ d 1≤s≤d ∥Sn ∥ bn n≽N (k) k=1 n̸≼N bn q ≤ C · 1≤s≤d ≥ϵ bn n≽N (k) k=1 ∥S n ∥ n̸≼N q bn ≥ϵ − E ∥S n ∥q q b(n,s,ns +1) − ξ (n) q b(n,s,ns +1) ck 1≼k≼n By (3.2), we have 1≤s≤d n ≽1 q bn − = 1≤s≤d ck k≽ ξ (n ) q b(n,s,ns +1) n ≽k bn ck ≼ k≼ n − q ξ (n) = q b(n,s,ns +1) 1≤s≤d cn · ψs (n) < ∞ n∈Nd Hence lim C · 1≤s≤d ∥N∥→∞ n̸≼N q bn − ξ (n) q b(n,s,ns +1) ck = 1≼k≼n and we are done The following theorem gives a characterization of p-uniformly smooth Banach spaces Theorem 3.2 Let ≤ p ≤ and E be a separable Banach space Then the following statements are equivalent: (i) E is of p-uniformly smooth (ii) Let {Xn , Fn ; n ∈ Nd } be a field of E-valued martingale differences, q ≥ 1, d ≥ 2, {bn , n ≽ 1} be a field positive constants such that bn ≤ bm for all n ≼ m If E ∥Sk ∥pq lim inf max = (3.3) pq ∥n∥→∞ n−1≺k≼n bk 1≤s≤d E ∥Xn ∥pq ϕs (n) < ∞, (3.4) n∈Nd where ϕs (n) = k≽ n pq bk − pq b(k,s,n +1) s |k|q−1 ; then Sn bn → strongly a.s as |n| → ∞ (3.5) Proof ((i) ⇒ (ii)) By Lemma 2.1 and Cr -inequality, we have q E ∥Sn ∥ pq ≤ CE ∥X k ∥ p ≤ C |n|q−1 k≼ n E ∥Xk ∥pq k≼n Applying Theorem 3.1 with ξ (n) = |n| , cn = E ∥Xn ∥pq , we have (ii) Now we prove [(ii) ⇒ (i)] Let {Yn1 , Gn1 ; n1 ≥ 1} be an arbitrary sequence of E-valued martingale differences such that q −1 ∞ E ∥Yn1 ∥p p n =1 n1 < ∞ T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 1907 For n = (n1 , , nd ) ∈ Nd Set if n1 ≥ 1, n2 = · · · = nd = otherwise Xn = Xn = Yn1 and Fn = σ {Xi ; i ≼ n} Then {Xn , Fn ; n ∈ Nd } is the field of E-valued martingale differences Let bn = |n|, q = 1, so Sn = ϕ1 ((n1 , 1, , 1)) = k ≥n 1≤s≤d E ∥Xn ∥p ϕs (n) ≤ n ≽1 E ∥Sn ∥p Moreover, − (k1 +1)p = p n1 p bn E ∥Xn ∥p ϕ1 (n) = n1 n1 nd i =1 E ∥Sn1 ∥p p n1 p p, n2 nd Xi1 → ∞ E ∥Yn1 ∥p i=1 Yi , < ∞ p n1 =1 n≽1 = n1 and p k1 n1 E ∥S ∥p so lim inf∥n∥→∞ maxn−1≺k≼n |kk|p = By (ii), a.s as |n| → ∞ Taking n2 = · · · = nd = and letting n1 → ∞, we obtain n1 n1 j = Yj → strongly a.s as n1 → ∞ Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier (1976), E is of p-uniformly smooth Remark It should be noted that for d = [(i) → (ii)] but [(ii) ̸→ (i)] Theorem 3.3 Let ≤ p ≤ 2, q > 1, E be a separable real p-uniformly smooth Banach space and {Xn , Fn ; n ∈ Nd } be a field of E-valued martingale differences If E ∥Xn ∥pq ϵ =o as |n| → ∞ for every ϵ > |n|1−α If {bn ; n ≽ 1} satisfying bn ≤ bm for all n ≪ m then P sup k̸≼n ∥S k ∥ bk >ϵ =o as |n| → ∞ for every ϵ > |n|1−α Proof Firstly, we show that P max ∥Sk ∥ > ϵ bn < ∞ k≼ n |n|α n ≽1 (3.8) Using Corollary 2.6 and Cr -inequality, we have |n|−α P max ∥ S ∥ > ϵ b ≤ C k n pq E k≼ n |n|α bn n ≽1 n ≽1 ≤C n∈Nd Then, we have (3.8) ∥X k ∥ bn E ∥Xk ∥pq p 1≼k≼n E ∥Xk ∥pq pq k≽ = |n|−α+p−1 n ≽1 =C q 1≼k≼n |n|−α+p−1 pq n ≽k bn E ∥Xn ∥q φ(n) < ∞ (by (3.7)) T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 1909 Next, with ϵ > 0, set An = {k, 2n−1 ≺ k ≼ 2n } If bn ≤ bm for all n ≼ m, we see n−α P sup b− k ∥Sk ∥ > ϵ = k≽ n n ≽1 i −1 2 n−α P sup b− k ∥Sk ∥ > ϵ k≽ n i≽1 n=2i−1 ≤C i −1 2 −i α |2 ≤C sup bk ∥Sk ∥ > ϵ |P k≽2i−1 |2i(1−α) |P sup max b− ∥ S ∥ > ϵ k k u≽i k∈Au i ≽1 |2 i(1−α) | i ≽1 ≤C P b− max ∥Sk ∥ > ϵ 2u−1 P n≽1 k≼2u u(1−α) |2 k≼2u |2i(1−α) | i≼u |P b2u maxu ∥Sk ∥ > −1 k≼2 u ≽1 ≤C b2u−1 max ∥Sk ∥ > ϵ −1 u ≽i u ≽1 ≤C −1 i≽1 n=2i−1 ≤C |n|−α P b− n max ∥Sk ∥ > k≼ n ϵ M ϵ M < ∞ (by (3.8)) d d Since {P supk≽n b− k ∥Sk ∥ > ϵ , n ∈ N } are non-increasing in n = (n1 , , nd ) for order relationship ≼ in N , it follows that P sup b− k ∥S k ∥ > ϵ =o k≽n |n|1−α as |n| → ∞ for all ϵ > the prove is the same as in the case (1) For bn = ∥n∥, n ≽ then bn ≤ bm for all n ≪ m For bn = |n|, n ≽ then bn ≤ bm for all n ≼ m Hence we obtain Corollary 3.7 Let ≤ p ≤ 2, q ≥ 1, α < and E be a real separable p-uniformly smooth Banach space, {Xn , Fn ; n ≽ 1} be a field of E-valued martingale differences If E ∥Xn ∥pq ϵ k̸≼n ∥k∥ =o |n|1−α as |n| → ∞ for every ϵ > and P sup k≽ n ∥S k ∥ >ϵ |k| =o |n|1−α as |n| → ∞ for every ϵ > Acknowledgments The authors would like to express their gratitude to the referee for his/her helpful comments and valuable suggestions which have significantly improved this paper The authors express their sincere thanks to the Advanced Math Program of Ministry of Education and Training, Viet Nam for sponsoring their working visit to University of Washington and to the Department of Mathematics, University of Washington for the hospitality This research was supported in part by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) 1910 T.C Son, D.H Thang / Statistics and Probability Letters 83 (2013) 1901–1910 References Brunk, H.D., 1948 The strong law of large numbers Duke Math J 15, 181–195 Christofides, T.C., Serfling, R.J., 1990 Maximal inequalities for multidimensionally indexed submartingale arrays Ann Probab 18 (2), 630–641 Fazekas, I., Klesov, O., 2000 A general approach to the strong law of large numbers Theory Probab Appl 45 (3), 436–449 Hoffmann-Jørgensen, J., Pisier, G., 1976 The law of large numbers and the central limit theorem in Banach spaces Ann Probab (4), 587–599 Hu, S., Chen, G., Wang, X., 2008 On extending the Brunk–Prokhorov strong law of large numbers for martingale differences Statist Probab Lett 78, 3187–3194 Lagodowski, Z.A., 2009 Strong laws of large numbers for B-valued random fields Discrete Dyn Nat Soc 12 http://dx.doi.org/10.1155/2009/485412 Article ID 485412 Noszaly, C., Tomacs, T., 2000 A general approach to strong laws of large numbers for fields of random variables Ann Univ Sci Budapest 43, 61–78 Prokhorov, Y.V., 1950 On the strong law of large numbers Bulletin the Soviet Union Academy of Sciences Ser Math 14, 523–536 (in Russian) Son, T.C., Thang, D.H., Dung, L.V., 2012 Rate of complete convergence for maximums of moving average sums of martingale difference fields in Banach spaces Statist Probab Lett 82, 1978–1985 Woyczyn’ski, W.A., 1978 Geometry and martingale in Banach spaces, part II, independent increments Adv Probab 4, 267–518 ... 1≼k≼n and we are done The following theorem gives a characterization of p-uniformly smooth Banach spaces Theorem 3.2 Let ≤ p ≤ and E be a separable Banach space Then the following statements are... 2000 A general approach to the strong law of large numbers Theory Probab Appl 45 (3), 436–449 Hoffmann-Jørgensen, J., Pisier, G., 1976 The law of large numbers and the central limit theorem in Banach. .. sponsoring their working visit to University of Washington and to the Department of Mathematics, University of Washington for the hospitality This research was supported in part by Vietnam’s National