Statistics and Probability Letters 80 (2010) 756–763 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Strong laws of large numbers for random fields in martingale type p Banach spaces Le Van Dung a,∗ , Nguyen Duy Tien b a Faculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Nam b Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam article abstract info Article history: Received 17 June 2009 Received in revised form 10 January 2010 Accepted 11 January 2010 Available online 29 January 2010 We extend Marcinkiewicz–Zygmund strong laws for random fields {Vn ; n ∈ Nd } with values in martingale type p Banach spaces Our results are more general and stronger than the result of Gut and Stadtmüller (2009) and some other ones © 2010 Elsevier B.V All rights reserved MSC: 60B11 60B12 60F15 60G42 Introduction Let Nd be the positive integer d-dimensional lattice points, where d is a positive integer For m = (m1 , , md ) and n = α (n1 , , nd ) ∈ Nd , notation m ≺ n means that mi ≤ ni , ≤ i ≤ d, |nα | is used for di=1 ni i , [m, n) = di=1 [mi , ni ) is a d-dimensional rectangle and i=1 (mi < ni ) means that there is at least one of m1 < n1 , m2 < n2 , ., md < nd holds We write = (1, , 1) ∈ Nd Consider a random field {Vn , n ∈ Nd } of random elements defined on a probability space (Ω , F , P ) taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space X with norm · In the current work, we establish strong laws of large numbers (SLLN) for |nα |−1 maxk≺n Sk This can be done by studying convergence of sums of type d |n|−1 P {max Sk > |nα |} for every > n k≺n Many authors have investigated the Marcinkiewicz type strong laws of large numbers for random fields {Xn , n ∈ Nd } of random variables For example, Fazekas and Tómács (1998) studied strong laws of large numbers |n|−1/r Sn (for some < r < 1) for pairwise independent random variables, Czerebak-Mrozowicz et al (2002) studied Marcinkiewicz type strong laws of large number |n|−1/p (Sn − ESn ) (for some < p < 2) for pairwise independent random fields Recently, Gut and Stadtmüller (2009) studied Marcinkiewicz–Zygmund laws of large numbers for random fields of i.i.d random variables In this paper, we not only extend these results to random fields in martingale type p Banach spaces but also bring more general and stronger ones 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 E-mail addresses: lvdunght@gmail.com (L.V Dung), nduytien2006@yahoo.com (N.D Tien) 0167-7152/$ – see front matter © 2010 Elsevier B.V All rights reserved doi:10.1016/j.spl.2010.01.007 L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 757 Preliminaries Technical definitions relevant to the current work will be discussed in this section Scalora (1961) introduced the idea of the conditional expectation of a random element in a Banach space For a random element V and sub-σ -algebra G of F , the conditional expectation E (V |G) is defined analogously to that in the random variable case and enjoys similar properties 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 p E Sn − Sn−1 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 p ∞ r E sup Sn ≤ CE Sn − Sn−1 n ≥1 p 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 ≤ then it is of martingale type r for all r ∈ [1, p) It follows from the Hoffmann-Jørgensen and Pisier (1976) 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 (1975, 1986) To prove the main result we need the following lemma which was proved by Dung et al (2009) in the case d = If d is arbitrary positive integer, then the proof is similar and so is omitted Lemma 2.1 Let ≤ p ≤ and let {Vk , k ≺ n} be a collection of |n| random elements in a real separable martingale type p Banach space with E (Vk |Fk ) = for all k ≺ n, where Fk is the σ -field generated by the family of random elements {Vl : di=1 (li < ki )}, F1 = {∅, Ω } Then p E max Sk ≤C k≺n E Vk p , k≺n where Sk = i ≺k Vi In the case of p = 1, the hypothesis that E (Vk |Fk ) = for all k ≺ n is superfluous Lemma 2.2 Let < p ≤ Let α1 , , αd be positive constants satisfying 1/p < min{α1 , αd } < 1, let q be the number of integers s such that αs = min{α1 , αd } If E V r (log+ V )q−1 < ∞ then we have ∞ (i) n | nα | |nα | n ≥ t }dt < ∞, |nα |p (ii) P{ V |nα |p P{ V p ≥ t }dt < ∞ Proof Without loss of generality, we may assume min{α1 , , αd } = α1 = · · · = αq < αq+1 ≤ αd We first prove (i) We have by Lemma of Stadtmüller and Thalmaier (2009) that g (j) = 1∼C αq+1 /α1 α /α 1≤n1 nq nq+1 nd d ≤j j(log j)q−1 as j → ∞ (q − 1)! Denote ∆g (j) = g (j) − g (j − 1) we get n |nα | ∞ ∞ |nα | P{ V ≥ t }dt ≤ k=1 ∞ ≤ k=1 ∞ ≤ k=1 kα1 kα1 kα1 ∞ ∞ ∆g (k) kα1 ∞ ∆g (k) i=k P{ V ≥ t }dt = k=1 (i+1)α1 i α1 P{ V iα1 −1 P { V i=k kα1 ≥ iα1 }dt ∞ ∆g (k) ≥ iα1 } ∞ ∆g (k) i=k (i+1)α1 iα1 P{ V ≥ t }dt 758 L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 ∞ ∞ ≤ k=1 ∞ i =k k=1 ∞ kα1 ∞ < (j + 1)α1 } jα1 P {jα1 ≤ V ∆g (k) j =k jα1 P {jα1 ≤ V < (j + 1)α1 } jα1 P {jα1 ≤ V < (j + 1)α1 } jα1 P {jα1 ≤ V < (j + 1)α1 } j =1 j j =1 ∞ =C j =1 ∆g (k) kα1 k=1 j ∞ =C iα1 −1 i=1 ∞ ≤C j < (j + 1)α1 } j =k k=1 < (j + 1)α1 } j =i P {jα1 ≤ V ∆g (k) kα1 ≤C P {jα1 ≤ V ∞ ≤ ∞ i α −1 ∆g (k) kα1 ∆g (k) kα1 k=1 j−1 k=1 kα1 − (k + 1)α1 − (k + 1)α1 g (k) ∞ P {jα1 ≤ V +C < (j + 1)α1 }g (j) j =1 ∞ jα1 P {jα1 ≤ V ≤C < (j + 1)α1 } j =1 j−1 k=1 kα1 k(log k)q−1 ∞ P {jα1 ≤ V < (j + 1)α1 }j(log j)q−1 jα1 P {jα1 ≤ V < (j + 1)α1 }(log j)q−1 +C j =1 ∞ ≤C j =1 j −1 k=1 kα1 − (k + 1)α1 ∞ P {jα1 ≤ V < (j + 1)α1 }j(log j)q−1 jα1 P {jα1 ≤ V < (j + 1)α1 }(log j)q−1 +C j =1 j ∞ ≤C j =1 kα1 k=1 ∞ P {jα1 ≤ V +C < (j + 1)α1 }j(log j)q−1 j =1 ∞ P {jα1 ≤ V ≤C < (j + 1)α1 }j(log j)q−1 < ∞ j =1 Now we prove (ii) |nα |p n |nα |p P{ V p 1 ≥ t }dt = | nα | p n ≤C n |nα |p P{ V p n n |nα |p |nα |p P{ V Noting that the first term on the right-hand side is finite, it remains to prove that |nα |p n |nα |p Denote d(k) = ∞ j=k d(j) jpα1 P{ V p ≥ t }dt n1 nq =k ∼C 1, we have by Lemma 3.1 of Gut (2001) that (log k)q−1 kpα1 −1 P{ V |nα |p +C |nα |p ≥ t }dt + p ≥ t }dt p ≥ t }dt k L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 759 Hence, we have n αq+1 |nα |p |nα |p ∞ P{ V p d(k) ≥ t }dt ≤ k,nq+1 , ,nd =1 α [kα1 nq+1 nd d ] pαq+1 E( V pαd kpα1 nq+1 nd p I (j ≤ V < j + 1)) j=1 (where [x] denotes the greatest integer not exceeding x) αq+1 ∞ d(k) ≤C k,nq+1 , ,nd =1 α [kα1 nq+1 nd d ] pαq+1 jp P (j ≤ V pαd kpα1 nq+1 nd αq+1 ∞ d(k) ≤C k,nq+1 , ,nd =1 d(k) k,nq+1 , ,nd =1 pαq+1 [jp − (j − 1)p ]P ( V ≥ j) pαd kpα1 nq+1 nd j =1 αq+1 α [kα1 nq+1 nd d ] pαq+1 pjp−1 P { V pαd kpα1 nq+1 nd =C d(k) ∞ pαq+1 pαd nq+1 , ,nd =1 nq+1 nd ≤C pαq+1 nq+1 , ,nd =1 nq+1 ∞ + pα nd d ≤C + ∞ ≤C nd which is finite if E ( V r kpα1 ≥ j} k= αq+1 α 1/α1 j/nq+1 nd d d(k) kpα1 α ∞ pj(1/α1 −1) P { V ≥ j} d(k) k=1 kpα1 ∞ ∞ pα nd d αq+1 α j=[nq+1 nd d ]+1 pjp−1 P { V ≥ j} k= αq+1 α 1/α1 j/nq+1 nd d d(k) kpα1 ∞ βq+1 nq+1 , ,nd =1 nq+1 d(k) ∞ pjp−1 P { V j =1 ∞ k=1 [nq+1 nd d ] βd pαq+1 nq+1 , ,nd =1 nq+1 ≥ j} pα nd d αq+1 α i=[nq+1 nd d ]+1 βq+1 nq+1 , ,nd =1 nq+1 ∞ pj(1/α1 −1) j(p−1/α1 ) P { V α ∞ αq+1 ∞ ≥ j} j =1 pαq+1 nq+1 , ,nd =1 nq+1 pjp−1 P { V j =1 [nq+1 nd d ] α [kα1 nq+1 nd d ] kpα1 k=1 αq+1 ∞ ≥ j} j =1 αq+1 ∞ α [kα1 nq+1 nd d ] ∞ ≤C < j + 1) j =1 βd nd jr −1 (log i)q−1 P { V ≥ j} j =1 log+ V )q−1 < ∞ and since βl = αl /α1 > for q + ≤ l ≤ d The random field {Vn , n ∈ Nd } is said to be weakly mean dominated by the random element V if, for some < C < ∞, |n| P { Vk ≥ x} ≤ CP { V ≥ x} k≺n for all n ∈ Nd and x > Main results With the preliminaries accounted for, the main results may now be established In the following, we let {Vn ; n ∈ Nd } be an array of random elements in a real separable Banach space X, Fk is the σ -field generated by the family of random d elements {Vl : i=1 (li < ki )}, F1 = {∅, Ω } The first theorem is a general a.s convergence one Theorem 3.1 Let α1 , , αd be positive constants Let {Vn , n ∈ Nd } be a random field of random elements If n |n| P {max Sk > |nα |} < ∞ k≺n for every > 0, (3.1) 760 L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 then |2nα | < ∞ for every P max Sl > l≺2n n >0 (3.2) and, a fortiori, the SLLN | nα | max |Sk | → a.s as |n| → ∞ (3.3) k≺n obtains Conversely, (3.2) implies that (3.1) holds > 0, denote 2n = (2n1 , , 2nd ) and 2nα = (2n1 α1 , , 2nd αd ) We have the inequalities Proof ((3.1) ⇒ (3.2)) Fix P max Sl > |2nα | ≤ l≺2n n k∈[2n ,2n+1 ) n P max Sl > l≺k ≤ k∈[2n ,2n+1 ) n |2n | 2d ≤ k∈[2n ,2n+1 ) n ≤ 2d n | n| |k| 2α1 +···,αd P max Sl > l ≺k P max Sl > l≺k P max Sl > l ≺n |kα | 2α1 +···+αd 2α1 +···+αd 2α1 +···+αd |kα | |kα | |nα | < ∞ (by (3.1)) This implies by the Borel–Cantelli lemma that |2nα | max Sk → a.s as |n| → ∞ k≺2n (3.4) Now for k ∈ [2n , 2n+1 ) we have 1 max Sl ≤ |kα | l≺k |kα | max l≺2n+1 Sl ≤ | 2nα max | l≺2n+1 Sl = 2α1 +···+αd |2(n+1)α | max l≺2n+1 and so the conclusion (3.3) follows from (3.4) and (3.5) ((3.2) ⇒ (3.1)) Suppose that (3.2) holds, we easily to prove that for every n |n| P max Sl > |nα | l ≺n P max Sl > ≤ n l≺2n 2α1 +···+αd Sl (3.5) > 0, |2nα | , which implies that (3.1) holds The proof is completed The following theorem characterizes the martingale type p Banach spaces Theorem 3.2 Let ≤ p ≤ and let X be a separable Banach space Then the following two statements are equivalent: (i) The Banach X is of martingale type p (ii) For every random field {Vn ; n ∈ Nd } in X with E (Vn |Fn ) = for all n ∈ Nd and for every α = (α1 , , αd ) with αi > for all ≤ i ≤ d, the condition E Vn n |nα |p p 0, P {max Sk > |nα |} < ∞ k≺n and, a fortiori, the SLLN |nα | obtains max Sk → k≺n a.s as |n| → ∞ L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 761 Proof In order to prove [(i) ⇒ (ii)] we show that P {max Sk > |2nα |} < ∞ > for every k≺2n n Applying Markov’s inequality and Lemma 2.2 we have that P {max Sk > |2nα |} ≤ p k≺2n n n |2nα |p E (max Sk ≤C n 2nα p | | p k≺2n E Vk k≺2n p ) E Vk ≤C k |kα |p p < ∞ Now we prove [(ii) ⇒ (i)] Assume that (ii) holds Let {Wn1 , Gn1 ; n1 ≥ 1} be an arbitrary sequence of martingale difference in X such that ∞ E Wn1 p p n1 n =1 |nα |} < ∞ (3.6) k≺n and, a fortiori, the SLLN |nα | max Sk → a.s as |n| → ∞ (3.7) k≺n obtains Proof For each n ∈ Nd , set Vk = Vk I ( Vk ≤ |nα |), Vk = Vk I ( Vk > |nα |), Yk = Vk − E (Vk |Fk ), Yk = Vk − E (Vk |Fk ), Sn = k≺n Yk , Sn = k≺n Yk Since E (Vk |Fk ) = 0, it follows that Vk = Yk + Yk Moreover, if Gk and Gk are the σ -fields generated by the family of random elements {Yl : i=1 (li < ki )} and {Yl : i=1 (li < ki )}, respectively, then Gk ⊂ Fk and Gk ⊂ Fk for all k ≺ n, which imply that E (Yk |Gk ) = E (Yk |Gk ) = for all k ≺ n d d 762 L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 > 0, We now begin the proof For every n |n| P {max Sk > |nα |} ≤ k≺n |n| n P {max Sk > |nα |} + k≺n |n| n > |nα |} P {max Sk k≺n (3.8) First, we show that n |n| P {max Sk k≺n > |nα |} < ∞ Applying Markov’s inequality and Lemma 2.2, we obtain n |n| P {max Sk k≺n > |nα |} ≤ |n||n|α n k≺n 1 ≤C |nα | |n| n ≤C |nα | n |nα | |n| =C |n| n ≤C k≺n n n 1 | nα | |n| |n| |nα | ∞ k≺n |nα | P { Vk ≥ t }dt P { Vk ≥ t }dt k≺n | nα | |nα | P{ V ≥ t }dt ∞ |nα | n |nα | ≥ t }dt ∞ ≥ |nα |} + C ∞ n k≺n n ≤C P { Vk k≺n P { Vk ≥ |nα |}dt + C P { Vk ≥ |nα |} + C P{ V ∞ |nα | |n| n k≺n 1 =C k≺n E Yk |nα | |n| n E Vk 1 E (max Sk ) ≤ C |nα | ≥ t }dt < ∞ (by Lemma 2.2) P{ V By (3.8), in order to complete the proof, we next show that n |n| P {max Sk > |nα |} < ∞ k≺n Again applying Markov’s inequality, we find that n |n| P {max Sk > |nα |} ≤ k≺n n |n| |nα |p n |n| |nα |p = 1 ≤C n k≺n k≺n n |nα |p n |nα |p k≺n k≺n |nα |p p ∞ |nα |p |n| E Yk k≺n P { Vk k≺n p ≥ t }dt P { Vk p ≥ t }dt P { Vk p ≥ t }dt |n| k≺n |nα |p ≤C =C |nα |p |n| |nα |p |nα |p |n| p n n =C n E Vk 1 E (max Sk > |nα |p ) ≤ C |nα |p |n| ≤C E (max Sk > |nα |)p P{ V p ≥ t }dt < ∞ (by Lemma 2.2) The proof is completed Remark Note that in the case of q < d, positive constants α1 , , αd are not upper bounded by 1, which is weaker than condition (2.1) of Theorem 2.1 of Gut and Stadtmüller (2009) Theorem 3.4 Let α1 , , αd be positive constants satisfying min{α1 , αd } > 1, let q be the number of integers s such that αs = min{α1 , αd } Suppose that {Vn , n ∈ Nd } is weakly mean dominated by V such that E V (log+ V )q−1 < ∞ Then (3.3) holds and then, the SLLN (3.4) obtains Proof The proof is similar to that of Theorem 3.2 with p = and we use Tn = Sn and Sn , respectively k≺n Vk and Tn = k≺n Vk are instead of L.V Dung, N.D Tien / Statistics and Probability Letters 80 (2010) 756–763 763 References Czerebak-Mrozowicz, E.B., Klesov, O.I., Rychlik, Z., 2002 Marcinkiewicz-type strong laws of large numbers for pairwise independent random fields Probab Math Statist 22 (Fasc 1), 127–139 Hoffmann-Jørgensen, J., Pisier, G., 1976 The law of large numbers and the central limit theorem in Banach spaces Ann Probab (4), 587–599 Fazekas, I, Tómács, T., 1998 Strong laws of large numbers for pairwise independent random variables with multidimensional indices Publ Math Debrecen 53 (1–2), 149–161 Dung, L.V., Ngamkham, Th., Tien, N.D., Volodin, A.I., 2009 Marcinkiewwcz-type law of large numbers for double arrays of random elements in Banach spaces Lobachevskii J Math 30 (4), 337–346 Gut, A., 2001 Convergence rates in the central limit theorem for multidimensionally indexed random variables Studia Sci Math Hungar 37, 401–418 Gut, A., Stadtmüller, U., 2009 An asymmetric Marcinkiewicz–Zygmund LLN for random fields Statist Probab Lett 79, 1016–1020 Pisier, G., 1975 Martingales with values in uniformly convex spaces Israel J Math 20 (3–4), 326–350 Pisier, G., 1986 Probabilistic methods in the geometry of Banach spaces In: Probability and Analysis (Varenna) In: Lecture Notes in Math., vol 1206 Springer, Berlin, pp 167–241 Scalora, F.S., 1961 Abstract martingale convergence theorems Pacific J Math 11, 347–374 Stadtmüller, U., Thalmaier, M., 2009 Strong laws for delayed sums of random fields Acta Sci Math (Szeged) 75 (3–4), 723–737 ... 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. .. Theorem 2.2 of Hoffmann-Jørgensen and Pisier (1976), X is of martingale type p In the next two theorems, we obtain the Marcinkiewicz–Zygmund type laws of large numbers for random fields of random. .. real separable Banach space of martingale type p for some < p ≤ then it is of martingale type r for all r ∈ [1, p) It follows from the Hoffmann-Jørgensen and Pisier (1976) characterization of Rademacher