DSpace at VNU: Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces

DSpace at VNU: Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces tài...

Georgian Math J 18 (2011), 777 – 800 DOI 10.1515/ GMJ.2011.0045 © de Gruyter 2011 Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces Nguyen Van Quang, Le Van Thanh and Nguyen Duy Tien Dedicated to Professor Nicholas Vakhania on the occasion of his 80th birthday Abstract For a double array of blockwise M-dependent random elements ¹Vmn W m 1; n 1º taking values in a real separable Rademacher type p (1 Ä p Ä 2) Banach space, we provide conditions to obtain the almost sure convergence for double sums Pm P n 1; n The paper treats two cases: (i) ¹Vmn W m 1; n 1º i D1 j D1 Vij , m is block-wise M-dependent with EVmn D 0, m; n 1, and (ii) ¹Vmn W m 1; n 1º is block-wise p-orthogonal The conditions for case (i) are shown to provide exact characterizations of Rademacher type p and stable type p Banach spaces Examples are given showing that the conditions cannot be removed or weakened It is also demonstrated that some of the well-known theorems in the literature are special cases of our results Keywords Blockwise M-dependent random elements, strong law of large numbers, double array of random elements, Rademacher type p Banach space, stable type p Banach space 2010 Mathematics Subject Classification 60F15, 60B11, 60B12 Introduction Móricz [15] introduced the concept of block-wise m-dependence for a sequence of random variables and extended the classical strong law of large numbers (SLLN) of Kolmogorov (see, e.g., Chow and Teicher [6, p 124]) to the block-wise mdependent case Móricz’s result [15] was extended by Gaposhkin [8] Based on a lemma of Chobanyan, Levental and Mandrekar [3], Rosalsky and Thanh [23] gave a simple proof of strong laws for sequences of block-wise m-dependent random elements in Banach spaces (see also [4, 7] for more details about this approach) The first and second author were supported in part by the National Foundation for Science and Technology Development, Vietnam (NAFOSTED), no 101.02.32.09 The third author was supported by the National Foundation for Science Technology Development, Vietnam (NAFOSTED), no 101032010.6 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 778 N V Quang, L V Thanh and N D Tien The SLLN for double arrays of block-wise independent random variables was also studied by Quang and Thanh [19] Recently, Móricz, Stadtmüller and Thalmaier [16] introduced the concept of block-wise M-dependence for a double array of random variables and established a double array version of the Kolmogorov SLLN for double arrays of random variables which are block-wise M-dependent with respect to the blocks ¹ Œ2k ; 2kC1 / Œ2l ; 2lC1 / W k 0; l 0º The results of Móricz, Stadtmüller and Thalmaier [16] were generalized by Stadtmüller and Thanh [27] In the present paper, we study this problem for double arrays of block-wise M-dependent random elements in Banach spaces Moreover, the conditions for the strong law of large numbers are shown to provide exact characterizations of Rademacher type p and stable type p Banach spaces The “asymmetric” Marcinkiewicz–Zygmund type SLLN for double arrays are also considered Some results in the literature, such as those in Gut [9], Gut and Stadtmüller [10], Móricz, Stadtmüller and Thalmaier [16], Móricz, Su and Taylor [17], Quang and Thanh [19], and Rosalsky and Thanh [21, 22] are improved and extended The following notation will be used throughout this paper For x 0, let Œx denote the greatest integer less than or equal to x For a; b R, min¹a; bº and max¹a; bº will be denoted, respectively, by a^b and a_b We use log to denote the logarithm to the base The symbol C denotes a generic constant (0 < C < 1) which is not necessarily the same one in each appearance The paper is organized as follows Technical definitions, notation, and the lemmas used in the proofs of the main results are presented in Section The main results are stated and proved in Sections and In Section 5, some examples are presented to illustrate the sharpness of the main results Preliminaries Some definitions, notation, and preliminary results will be presented prior to establishing the main results Let X be a real separable Banach space with norm k k A random element in X will be denoted by V or by Vmn , etc The expected value or mean of a random element V , denoted by EV , is defined to be the Pettis integral provided it exists That is, V has an expected value EV X if f EV / D E.f V // for every f X where X denotes the (dual) space of all continuous linear functionals on X (see, e.g., Vakhania, Tarieladze and Chobanyan [30, p 113]) If EkV k < 1, then (see, e.g., Taylor [28, p 40]) V has an expected value But the expected value can exist when EkV k D 1, see, e.g., Taylor [28, p 41] Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM Almost sure convergence for double arrays in Banach spaces 779 A double array of random elements ¹Vmn W m 1; n 1º is said to be stochastically dominated by a random element V if for some constant C < P ¹kVmn k > tº Ä CP ¹kV k > tº; t 0; m 1; n 1: (2.1) For a double array ¹Vmn W m 1; n 1º of identically distributed random elements, this condition will automatically hold with V D V11 and C D It follows from Lemma 5.2.2 of [28, p 123] that stochastic dominance can be accomplished by the random elements ¹Vmn W m 1; n 1º having a bounded absolute rth moment (r > 0) Specifically, if X EkVmn kr < 1; m 1;n then there exists a random element V such that EkV kp < for all < p < r and (2.1) holds with C D (The condition r > of [28, Lemma 5.2.2] is not needed, as was pointed out by Adler, Rosalsky and Taylor [2]) Let ¹Yn W n 1º be a symmetric Bernoulli sequence; that is, ¹Yn W n 1º be a sequence of independent and identically distributed (i i d.) random variables with P ¹Y1 D 1º D P ¹Y1 D 1º D 1=2 Let X D X X X and define ° ± X C.X/ D v1 ; v2 ; : : : / X W Yn converges in probability : nD1 Let Ä p Ä Then X is said to be of Rademacher type p if there exists a constant < C < such that E X nD1 p Y n ÄC X kvn kp for all v1 ; v2 ; : : : / C.X/: nD1 Hoffmann–Jørgensen and Pisier [11] proved for Ä p Ä that a real separable Banach space is of Rademacher type p if and only if there exists a constant < C < such that n n X X p ÄC EkVi kp (2.2) Vi E i D1 i D1 for every finite collection ¹V1 ; : : : ; Vn º of independent mean random elements If a real separable Banach space is of Rademacher type p for some < p Ä 2, then it is of Rademacher type q for all Ä q < p Every real separable Banach space is of Rademacher type (at least) 1, while the Lp - and `p -spaces are of Rademacher type ^ p for p Every real separable Hilbert space and real Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 780 N V Quang, L V Thanh and N D Tien separable finite-dimensional Banach space is of Rademacher type In particular, the real line R is of Rademacher type Let < p Ä and let ¹Ân W n 1º be independent and identically distributed stable random variables, each with E exp.i t /P D exp¹ jtjp º The separable Banach space X is said to of stable type p if nD1 Ân converges a.s Pbe p < whenever X, n with kv k n nD1 Equivalent characterizations of a Banach space being of stable type p, properties of stable type p Banach spaces, as well as various relationships between the conditions Rademacher type p and stable type p may be found in [1,14,18,25,31] Some of these properties and relationships will now be summarized (i) Every separable Banach space X is of stable type p for all < p < (ii) For q 2, the Lq -spaces and lq -spaces are of stable type 2, while for Ä q < 2, the Lq -spaces and lq -spaces are of stable type p for all < p < q but are not of stable type q (iii) Every separable Hilbert space and separable finite dimensional Banach space is of stable type (iv) For Ä p < 2, X is of stable type p if and only if X is of Rademacher type p1 for some p1 pI 2 (v) For p D 2, X is of stable type if and only if X is of Rademacher type The concept of block-wise M-dependence was introduced by Móricz, Stadtmüller and Thalmaier [16] and by Stadtmüller and Thanh [27] as follows Let M be a nonnegative integer A finite collection of random elements ¹V11 ; : : : ; Vmn º is said to be M-dependent if either m _ n Ä M C or m _ n > M C and the random elements ¹V11 ; : : : ; Vij º are independent of the random elements ¹Vkl ; : : : ; Vmn º whenever k i/ _ l j / > M A double array of random variables ¹Vmn W m 1; n 1º is said to be M-dependent if for each m 1; n 1, the random elements ¹V11 ; : : : ; Vmn º are M-dependent The notion of p-orthogonality of random elements was introduced by Howell and Taylor [12], and by Móricz, Su and Taylor [17] A finite collection of random elements ¹V11 ; : : : ; Vmn º is said to be p-orthogonal (1 Ä p < 1) if EkVij kp < for all Ä i Ä m, Ä j Ä n and E j i X X uD1 vD1 p a u/ v/ V u/ v/ ÄE k X l X p a u/ v/ V u/ v/ uD1 vD1 for all choices of Ä i Ä k Ä m, Ä j Ä l Ä n for all constants ¹a11 ; : : : ; akl º, and for all permutations and of the integers ¹1; : : : ; kº and ¹1; : : : ; lº, respectively An array of random elements ¹Vmn W m 1; n 1º is said to be Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 781 Almost sure convergence for double arrays in Banach spaces p-orthogonal (1 Ä p < 1) if ¹V11 ; : : : ; Vmn º is p-orthogonal for all m ^ n We refer to Howell and Taylor [12] and Móricz, Su and Taylor [17] for a detailed discussion of p-orthogonality Móricz, Su and Taylor [17] also established the Rademacher–Menshov type SLLN for double arrays of p-orthogonal random elements Chobanyan and Mandrekar [5] and Quang and Thanh [20] studied the SLLN problem for p-orthogonal random elements under rearrangements Let ¹!.k/ W k 1º and ¹ k/ W k 1º be strictly increasing sequences of positive integers with !.1/ D 1/ D and set kl D !.k/; !.k C 1/ l/; l C 1/ : We say that an array ¹Vij W i 1; j 1º of random elements is block-wise M-dependent (resp., block-wise p-orthogonal (1 Ä p < 1)) with respect to the blocks ¹kl W k 1; l 1º, if for each k and l, the random variables ¹Vij W i; j / kl º are M-dependent (resp., p-orthogonal) Thus the random elements with indices in each block are M-dependent (resp., p-orthogonal) but there are no M-dependence (resp., p-orthogonality) requirements between the random elements with indices in different blocks; even repetitions are permitted For ¹!.k/ W k 1º, ¹ k/ W k 1º and ¹kl W k 1; l 1º as above, and for m 0; n 0; k 1; l 1, we introduce the following notation: ® ¯ .mn/ D i; j / W 2m Ä i < 2mC1 ; 2n Ä j < 2nC1 ; mn/ kl Imn m/ rk n/ sl D kl \ .mn/ ; ¯ ® mn/ D k; l/ W kl 6D ; ; ® ¯ D i W i Œ!.k/; !.k C 1// \ Œ2m ; 2mC1 / ; ¯ ® D i W j Œ l/; !.l C 1// \ Œ2n ; 2nC1 / ; cmn D card Imn ; dmn D '.k; l/ D max mn/ k;l/2Imn X X card kl ; cij I.ij / k; l/; i D0 j D0 k; l/ D X X dij I.ij / k; l/; i D0 j D0 k; l/ D max '.i; j /; i Äk;j Äl where I.ij / denotes the indicator function of the set .ij / , i 0, j Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 782 N V Quang, L V Thanh and N D Tien In [19], the following relations are verified (i) If !k D k D 2k 1, k 1, then cmn D 1; '.m; n/ D 1; (ii) If !k D k D bq k 1c ˛ m; n/ D 1; 1; n 1: (2.3) m; n/ D O.1/: (2.4) D b2k c for all large k where < ˛; ˇ < 1, then k D bk ˇ c, k ˛/=˛ ˇ /=ˇ n D k, k ˛/=˛ /; Log n/.1 ˇ /=ˇ (2.5) /: where ˛ > 1; ˇ > 1, then cmn D O.2m=˛ 2m=ˇ / and k m ˇ k '.m; n/ D O Log m/.1 (v) If ˇk D and '.m; n/ D O.1/; cmn D O.m.1 (iv) If !k D bk ˛ c, 0; n for all large k where q > 1, then cmn D O.1/ (iii) If !k D b2k c, m '.m; n/ D O.m1=˛ n1=ˇ /: (2.6) 1, then cmn D 2mCn ; m; n and '.m; n/ D O.mn/: (2.7) The following lemma establishes the maximal inequality for double sums of independent random variables which is due to Rosalsky and Thanh [21] Lemma 2.1 Let ¹Vij W Ä i Ä m; Ä j Ä nº be a double array of independent mean random elements in a real separable Rademacher type p (1 Ä p Ä 2) Banach space Then E where Skl D Pk i D1 max kSkl k 1ÄkÄm 1ÄlÄn Pl j D1 Vij , p Á ÄC m X n X EkVij kp ; (2.8) i D1 j D1 C is a constant independent of m and n Using the method of Móricz, Stadtmüller and Thalmaier [16] and Lemma 2.1, we can establish the maximal inequality for double sums of M-dependent random elements In the proof of Lemma 2.2, we sometimes denote Vij by V iI j / for convenience Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 783 Almost sure convergence for double arrays in Banach spaces Lemma 2.2 Let ¹Vij W Ä i Ä m; Ä j Ä nº be a double array of M-dependent mean random elements in a real separable Rademacher type p (1 Ä p Ä 2) Banach space Then E where Skl D Pk iD1 m X n Á X max kSkl kp Ä C EkVij kp ; 1ÄkÄm 1ÄlÄn Pl j D1 Vij , (2.9) i D1 j D1 C is a constant independent of m and n Proof If m _ n Ä M C 1, Lemma 2.2 is trivial So, we only need to consider m _ n M C In the case p D 1, note that for all m and n E l k X X max 1ÄkÄm 1ÄlÄn i D1 j D1 Vij Á ÄE DE max l k X X kVij k Á 1ÄkÄm 1ÄlÄn i D1 j D1 m X n X m X n Á X kVij k D EkVij k; i D1 j D1 iD1 j D1 establishing (2.9) In the case < p Ä 2, if m ^ n > M C 1, then E max k X l X 1ÄkÄm 1ÄlÄn i D1 j D1 ÄE C1 M C1 M X X i D1 j D1 ÄC M C1 X E i;j D1 ÄC pÁ Vij k X l X max 0Äk.M C1/Äm i 0Äl.M C1/Än j 0Äk.M C1/Äm i 0Äl.M C1/Än j M C1 M C1 X X i D1 j D1 X E V u.M C 1/ C i I v.M C 1/ C j DC pÁ uD0 vD0 X 0Äk.M C1/Äm i 0Äl.M C1/Än j kV k.M C 1/ C i I l.M C 1/ C j kp m X n X pÁ uD0 vD0 k X l X max V u.M C 1/ C i I v.M C 1/ C j Á (by Lemma 2.1) EkVij kp ; (2.10) i D1 j D1 again establishing (2.9) If m ^ n Ä M C 1, the proof is similar to (2.10) Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 784 N V Quang, L V Thanh and N D Tien The next lemma is a Rademacher–Menshov maximal inequality for double arrays of p-orthogonal random elements in Banach spaces For a proof see Móricz, Su and Taylor [17] Lemma 2.3 Let ¹Vij W Ä i Ä m; Ä j Ä nº be a double array of random elements in a real separable Rademacher type p (1 Ä p Ä 2) Banach space If ¹Vij W Ä i Ä m; Ä j Ä nº is p-orthogonal, then E m X n Á X EkVij kp log m/p log n/p ; max kSkl kp Ä C 1ÄkÄm 1ÄlÄn where Skl D Pk iD1 (2.11) i D1 j D1 Pl j D1 Vij , C is a constant independent of m and n The following lemma establishes the strong law of large numbers for double arrays of arbitrary random elements It is based on Thanh [29, Theorem 3.1] and the remark thereafter concerning the case < p Ä Lemma 2.4 Let ¹Vij W i let ˛ > 0; ˇ > If 1; j 1º be a double array of random elements and 1 X X EkVij kp 1; a2n inf n and the condition a2nC1 < 1; a2n b2nC1 > 1; b2n (3.1) b2nC1 < 1; b2n (3.2) inf n sup sup n n X X EkVij kp < 1; bj /p (3.3) i D1 j D1 implies m_n!1 am bn m; n//.p lim m X n X 1/=p Vij D a.s (3.4) i D1 j D1 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 786 N V Quang, L V Thanh and N D Tien Proof We first prove the implication (i) ) (ii) Set mn/ Tkl D v X u X max mn/ u;v/2kl k; l/ Imn ; m Vij ; 0; n 0; n/ m/ i Drk j Dsl and mn/ P Tmn D For m Tkl k;l/2Imn a2mC1 0, n E.Tmn /p Ä 1/=p ; m 0; n 0: C a2mC1 b2nC1 /p 2m ; 2n //p C a2mC1 b2nC1 /p D Ä b2n / .2m ; 2n //.p a2m /.b2nC1 C a2mC1 b2nC1 /p cp 1 mn X mn/ p E.Tkl / (by (3.1)) k;l/2Imn mn/ p X E.Tkl / k;l/2Imn X X EkVij kp (by Lemma 2.2) k;l/2Imn i;j /2.mn/ kl C a2mC1 b2nC1 /p D ÄC X i;j /2.mn/ nC1 2mC1 X 12X1 i D2m EkVij kp j D2n EkVij kp : bj /p P P1 p It follows from (3.3) that i D1 j D1 E.Tij / < and so by the Markov inequality and the Borel–Cantelli lemma Tmn ! Note that for m 2l Ä n < 2lC1 , 1, n 1, letting k Pm Pn 0Ä j D1 Vij m; n//.p 1/=p i D1 am bn a.s as m _ n ! 1: Ä 0, l k X l X a2iC1 i D0 j D0 (3.5) be such that 2k Ä m < 2kC1 , a2i /.b2j C1 a2k b2l b2j / Tij : (3.6) Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 787 Almost sure convergence for double arrays in Banach spaces It follows from (3.2) that k X l X a2iC1 i D0 j D0 a2i /.b2j C1 a2k b2l b2j / 0, ˇ > 0, reduces to a result of Rosalsky and Thanh [22], and Móricz, Su and Taylor [17] by taking kl D 2k ; 2k 2l ; 2l ; k 1; l and recalling (2.3) Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 789 Almost sure convergence for double arrays in Banach spaces Theorem 3.3 Let ¹Vmn W m 1; n 1º be a double array of random elements in a real separable Rademacher type p Ä p Ä 2/ Banach space and let ¹an W n 1º and ¹bn W n 1º be nondecreasing sequences of positive constants satisfying (3.1) and (3.2) If ¹Vmn W m 1; n 1º is block-wise p-orthogonal with respect to the blocks ¹kl W k 1; l 1º and if X X EkVij kp log i /p log j /p < 1; bj /p (3.11) i D1 j D1 then n m X X m_n!1 am bn m; n//.p lim 1/=p Vij D (3.12) a.s i D1 j D1 Proof We first prove the implication (i) ) (ii) Set mn/ Tkl D max u X v X k; l/ Imn ; m Vij ; mn/ u;v/2kl 0; n m/ n/ i Drk j Dsl and mn/ P Tmn D For m k;l/2Imn a2mC1 0, n E.Tmn /p Ä D Ä a2m /.b2nC1 Tkl b2n / .2m ; 2n //.p 1/=p m 0; n 0: C p a2mC1 b2nC1 / 2m ; 2n //p C a2mC1 b2nC1 /p C a2mC1 b2nC1 /p X cp 1 mn X C a2mC1 b2nC1 /p ÄC j D2n / (by (3.1)) mn/ p E.Tkl / k;l/2Imn X X EkVij kp log 2m /p log 2n /p k;l/2Imn i;j /2.mn/ X (by Lemma 2.3) EkVij kp log 2m /p log 2n /p i;j /2.mn/ nC1 2mC1 X 12X1 i D2m mn/ p E.Tkl k;l/2Imn kl D ; EkVij kp log i /p log j /p : bj /p The rest of the argument is exactly the same as that used to complete the proof of the implication (i) ) (ii) of Theorem 3.1 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 790 N V Quang, L V Thanh and N D Tien Strong laws for the stochastic dominance case In this section, we establish SLLN for double arrays of block-wise M-dependent random elements which are stochastically dominated by a random variable As we have mentioned in Section 2, this case is an extension of the identically distributed case The following theorem extends Corollary 3.1 of Rosalsky and Thanh [22] to the block-wise M-dependent case Therefore, it extends the SLLN for double arrays of independent identically distributed real valued random variables which are established by Smythe [26] and Gut [9] Theorem 4.1 Let Ä p < 2, X be a real separable Rademacher type q Banach space for some p < q Ä Let ¹Vmn W m 1; n 1º be a double array of random elements which is stochastically dominated by a random element V If ¹Vmn W m 1; n 1º is block-wise M-dependent with respect to some array of blocks ¹kl W k 1; l 1º, and if E.kV kp logC kV k/ < 1; (4.1) then 1=p m_n!1 mn/ m; n//.q lim m X n X 1/=q Vij EVij / D a.s (4.2) i D1 j D1 Proof Set Uij D Vij I.kVij k Ä ij /1=p /; i 1; j 1; Wij D Vij I.kVij k > ij /1=p /; i 1; j 1: By a standard computation, we get and X X EkUij EUij kq < 1; q=p ij / i D1 j D1 (4.3) X X EkWij EWij k < if p > 1: 1=p ij / i D1 j D1 (4.4) By applying Theorem 3.1 with an D bn D n1=p ; n 1=p m_n!1 mn/ m; n//.q lim m X n X 1/=q Uij 1, it follows from (4.3) that EUij / D a.s (4.5) i D1 j D1 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 791 Almost sure convergence for double arrays in Banach spaces If p D 1, then X X P ¹Uij 6D Vij º D i D1 j D1 X X P ¹kVij k > ij º i D1 j D1 ÄC X X P ¹kV k > ij º (by (2.1)) i D1 j D1 ij // ÄC mn i D1 j D1 !0 as m _ n ! 1: (4.7) It follows from (4.5) (with p D 1), (4.6) and (4.7) that m X n X Vij m_n!1 mn .m; n//1=2 i D1 j D1 lim EVij / D If < p < 2, by applying Lemma 2.4 with an D bn D n1=p , n from (4.4) that m X n X Wij m_n!1 mn/1=r i D1 j D1 lim EWij / D a.s a.s 1, it follows (4.8) The conclusion (4.2) follows immediately from (4.5) and (4.8) Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 792 N V Quang, L V Thanh and N D Tien If m and n in Theorem 3.3 take different powers, condition (4.1) can be replaced by the weaker condition EkV kp < 1: (4.9) This problem for independent identically distributed real valued random variables was considered by Gut and Stadtmüller [10] and they referred to it as “asymmetric” Marcinkiewicz–Zygmund SLLN The following theorem reduces to Theorem 2.1 of Gut and Stadtmüller [10] when X is the real line and the random variables are independent (it should be noted that Gut and Stadtmüller [10] considered the d -dimensional case (d 2)) Moreover, this theorem provides a new characterization of stable type p Banach spaces in terms of the strong law of large numbers Theorem 4.2 Let X be a real separable Banach space and let Ä r < p < Then the following statements are equivalent (i) X is of stable type p (ii) For every double array ¹Vmn W m 1; n 1º of random elements in X which is stochastically dominated by a random element V and block-wise M-dependent with respect to some array of blocks ¹kl W k 1; l 1º, condition (4.9) implies Pm Pn EVij / i D1 j D1 Vij 1=p 1=r q m_n!1 m n m; n// 1/=q lim D0 a.s for some q > p: (4.10) (iii) For every double array ¹Vmn W m 1; n 1º of independent symmetric random elements in X which is stochastically dominated by a random element V , condition (4.9) implies m X n X Vij D m_n!1 m1=p n1=r i D1 j D1 lim a.s (4.11) Proof (i) ) (ii) Assume that X is of stable type p Then X is of Rademacher type q for some p < q < Set Uij D Vij I.kVij k Ä i 1=p j 1=r /; i 1; j 1; Wij D Vij I.kVij k > i 1=p j 1=r /; i 1; j 1: and Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 793 Almost sure convergence for double arrays in Banach spaces Firstly, X X EkUmn EUmn kq m1=p n1=r /q mD1 nD1 Ä2 Ä2 X X EkUmn kq m1=p n1=r /q mD1 nD1 X X 1=p n1=r /q m mD1 nD1 m1=p n1=r Z qx q P ¹kVmn k > xºdx X X Z m1=p n1=r ÄC qx q P ¹kV k > xºdx 1=p n1=r /q m mD1 nD1 X Z X ® ¯ DC P kV k > t 1=q m1=p n1=r dt mD1 nD1 Z 1 X ÂX ±Ãà ° kV k 1=p DC P 1=q 1=r > m dt t n nD1 mD1 à Z 1 1 X p D CE.kV k / dt < 1: t p=q nD1 np=r By applying Theorem 3.1 with an D n1=p , bn D n1=r , n (4.12) that lim m_n!1 m1=p n1=r m; n//.q m X n X 1/=q Uij (4.12) 1, it follows from EUij / D a.s (4.13) i D1 j D1 Next, X X EkWmn EWmn k m1=p n1=r mD1 nD1 Ä2 X X EkWmn k m1=p n1=r mD1 nD1 X X D2 1=p m n1=r mD1 nD1 Z P ¹kWmn k > xºdx Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 794 N V Quang, L V Thanh and N D Tien X X D2 1=p m n1=r mD1 nD1 m1=q n1=r Z X X C2 1=p m n1=r mD1 nD1 X X ÄC 1=p n1=r m mD1 nD1 Z nD1 mD1 CC P ° kV k n1=r mD1 nD1 1 P ¹kVmn k > xºdx ® ¯ P kV k > m1=p n1=r dx > m1=p X Z X m1=p n1=r m1=p n1=r CC 1=p m n1=r mD1 nD1 1 X X Z X X DC ® ¯ P kVmn k > m1=p n1=r dx Z m1=p n1=r P ¹kV k > xºdx ± ® ¯ P kV k > tm1=p n1=r dt Z 1 X X 1 ° kV k ±Ã X EkV kp 1=p CC P 1=r > m dt ÄC np=r n t nD1 mD1 nD1 à Z 1 X EkV kp dt ÄC CC np=r t p nD1 ÂX ÃZ 1 1 p Ä C C CEkV k dt < 1: (4.14) p=r n nD1 By Lemma 2.4 with ˛ D 1=p, ˇ D 1=r, it follows from (4.14) that m X n X Zij m_n!1 m1=p n1=r i D1 j D1 lim EZij / D a.s (4.15) The conclusion (4.10) follows immediately from (4.13) and (4.15) (ii) ) (iii) (iii) follows immediately from (ii) by virtue of (2.3) (iii) ) (i) Let ¹Wk W k 1º be a double array of independent symmetric random elements in X which is stochastically dominated by a random element V with EkV kp < For m 1, n 1, set ´ Wm if n D 1; Vmn D if n 2: Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 795 Almost sure convergence for double arrays in Banach spaces Then ¹Vmn W m 1; n 1º is a double array of independent symmetric random elements in X which is stochastically dominated by V Now by (iii), Pm Pm i D1 Wi i D1 Vi1 lim D lim m_n!1 m1=p n1=r m_n!1 m1=p n1=r Pm Pn i D1 j D1 Vij D lim D a.s 1=p m_n!1 m n1=r It then follows, by taking n D and letting m ! 1, that Pm i D1 Wi lim D a.s.; (4.16) m!1 m1=p and so by Lemma 2.5, X is of stable type p Some examples We conclude by presenting three illustrative examples The first example illustrates Theorem 3.1 In this example, it is shown that we can apply our result to obtain the SLLN while [21, Theorem 3.1] cannot be applied Example 5.1 Let ¹Wij W i 1; j 1º be a double array of independent mean random elements in a real separable Rademacher type p (1 Ä p Ä 2) Banach space and suppose that Wij is not independent of itself and X X EkWij kp 0, ˇ > 0: (5.1) Let Vij D W.i 2k C1/;.j 2l C1/ ; 2k Ä i < 2kC1 ; 2l Ä j < 2lC1 ; k Then EVmn D 0, m 1, n 1, and ¹Vmn W m 1; n 0-dependent with respect to the blocks ® k kC1 ¯ ;2 2l ; 2lC1 W k 0; l : 0; l 0: 1º is block-wise Now, using (5.1) we obtain kC1 lC1 X 1 X X 2X1 X X EkVij kp EkW i D ˛p ˇp i j k l i D1 j D1 kD0 lD0 i D2 2k D j D2 2k C 1I j i ˛p j ˇp 2l C 1/kp 2l X XX X kD0 lD0 EkW i I j /kp i C 2k 1/˛p j C 2l i D1 j D1 1/ˇp Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 796 N V Quang, L V Thanh and N D Tien D X X X X i D1 j D1 kDŒlog i lDŒlog j  Ä X X X X i D1 j D1 kDŒlog i lDŒlog j  ÄC EkW i I j /kp i C 2k 1/˛p j C 2l 1/ˇp EkW iI j /kp 2k˛p 2lˇp X X EkW i I j /kp < 1: ˛p j ˇp i i D1 j D1 (5.2) Thus, by (2.3) and Theorem 3.1, m n XX Vij D m_n!1 m˛ nˇ i D1 j D1 lim a.s (5.3) However, [21, Theorem 3.1] cannot be applied because the array ¹Vmn W m n 1º is not independent 1; The following example, which is a modification of Example 4.4 of Rosalsky and Thanh [22], shows apropos the implication (i) ) (ii) in Theorem 3.1 that under its hypotheses the series n m X X Vij =.ai bj / i D1 j D1 can diverge a.s as m ^ n ! Consequently, the conclusion of Theorem 3.1 cannot in general be reached through the well-known Kronecker lemma approach for proving SLLN as was done in Smythe [26] and Gut [9] Example 5.2 Let the underlying Banach space be the real line and let p D Let ¹Xn W n 1º be a sequence of independent mean random variables such that P1 P ạX1 Ô 0º D and nD1 EXn2 < Define for n ´ nX1 if Log n N; Wn D 1CLog n nXn if Log n … N; and let an D bn D n, n 1, and ´ Vmn D Wn if m D 1; if m 2: Then EVmn D 0, m 1, n 1, and ¹Vmn W m 1; n 0-dependent with respect to the blocks ® k kC1 ¯ ;2 2l ; 2lC1 W k 0; l : 1º is block-wise Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM Almost sure convergence for double arrays in Banach spaces Now 797 1 X X X EWn2 D EX C EXn2 < 1; bn2 C m/2 nD1 mD0 nD3 Log n…N and so (3.3) holds by recalling (2.3) However, n X i D1 Log i 2N bLog nc X X1 D X1 C Log i 1Cm diverges a.s as n ! 1; mD0 and by the Khintchine–Kolmogorov convergence theorem (see, e.g., Chow and Teicher [6, p 113]) n X Xi converges a.s as n ! 1: i D3 Log i …N Consequently, for m 1, n m X n n X X Vij Wj D bj bj i D1 j D1 j D1 D n X i D1 Log i 2N n X X1 Xi C C Log i diverges a.s as m ^ n ! 1: i D3 Log i …N Finally, we recall Example 4.1 in Rosalsky and Thanh [24] We will show by this example that in Theorem 4.1, the hypothesis that X is of Rademacher type q for some q > p (or, equivalently, that X is of stable type p) cannot be weakened to the hypothesis that X is of Rademacher type p Example 5.3 For Ä p < 2, consider the real separable Banach space `p consisting of absolutely pth power summable real sequences v D ¹vk ; k 1º with P p /1=p It is known that ` is of Rademacher type p, but jv j norm kvk D p kD1 k it is not of Rademacher type q for q > p The element having in its kth position and elsewhere will be denoted by v k/ , k Let ' W N N ! N be a oneto-one and onto map Let ¹Vmn W m 1; n 1º be a double array of independent random variables in `p by requiring the ¹Vmn W m 1; n 1º to be independent with ® ¯ ® ¯ P Vmn D v ' m;n/// D P Vmn D v ' m;n/// D ; m 1; n 1: Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 798 N V Quang, L V Thanh and N D Tien Then the array ¹Vmn W m 1; n 1º is stochastically dominated by V11 with EkV11 kp < Moreover, since ¹Vmn W m 1; n 1º is block-wise 0-dependent with respect to the blocks ® 2k ; 2k 2l ; 2l W k 1; l ¯ ; by calling (2.3), we get m; n/ Á On the other hand Pm Pn Pm Pn EVij / mn/1=p i D1 j D1 Vij i D1 j D1 Vij D D D1 mn/1=p mn/1=p mn/1=p a.s Therefore (4.2) fails Bibliography [1] A Adler, M O Cabrera, A Rosalsky and A I Volodin, Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces, Bull Inst Math Acad Sinica 27 (1999), no 3, 187–212 [2] A Adler, A Rosalsky and R L Taylor, Some strong laws of large numbers for sums of random elements, Bull Inst Math Acad Sinica 20 (1992), no 4, 335–357 [3] S Chobanyan, S Levental and V Mandrekar, Prokhorov blocks and strong law of large numbers under rearrangements, J Theoret Probab 17 (2004), no 3, 647–672 [4] S Chobanyan, S Levental and H Salehi, Strong law of large numbers under a general moment condition, Electron Comm Probab 10 (2005), 218–222 [5] S Chobanyan and V Mandrekar, On Kolmogorov SLLN under rearrangements for “orthogonal” random variables in a B-space, J Theoret Probab 13 (2000), no 1, 135–139 [6] Y S Chow and H Teicher, Probability Theory Independence, Interchangeability, Martingales, 3rd ed., Springer Texts Statist., Springer, New York, 1997 [7] I Fazekas and O Klesov, A general approach to the strong laws of large numbers (in Russian), Teor Veroyatnost i Primenen 45 (2000), no 3, 568–583; translation in Theory Probab Appl 45 (2002), no 3, 436–449 [8] V F Gaposhkin, On the strong law of large numbers for blockwise-independent and blockwise-orthogonal random variables (in Russian), Teor Veroyatnost i Primenen 39 (1994), no 4, 804–812; translation in Theory Probab Appl 39 (1994), no 4, 677–684 [9] A Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann Probability (1978), no 3, 469–482 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM Almost sure convergence for double arrays in Banach spaces 799 [10] A Gut and U Stadtmüller, An asymmetric Marcinkiewicz–Zygmund LLN for random fields, Statist Probab Lett 79 (2009), no 8, 1016–1020 [11] J Hoffmann-Jørgensen and G Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann Probability (1976), no 4, 587–599 [12] J O Howell and R L Taylor, Marcinkiewicz-Zygmund weak laws of large numbers for unconditional random elements in Banach spaces, in: Probability in Banach Spaces, III (Medford, 1980), Lecture Notes in Math 860, Springer, Berlin (1981), 219–230 [13] M Loève, Probability Theory, I, 4th ed., Grad Texts in Math 45, Springer, New York, 1977 [14] M B Marcus and W A Woyczy´nski, Stable measures and central limit theorems in spaces of stable type, Trans Amer Math Soc 251 (1979), 71–102 [15] F Móricz, Strong limit theorems for block-wise m-dependent and block-wise quasiorthogonal sequences of random variables, Proc Amer Math Soc 101 (1987), no 4, 709–715 [16] F Móricz, U Stadtmüller and M Thalmaier, Strong laws for block-wise M-dependent random fields, J Theoret Probab 21 (2008), no 3, 660–671 [17] F Móricz, K L Su and R L Taylor, Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces, Acta Math Hungar 65 (1994), no 1, 1–16 [18] G Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math 1206, Springer, Berlin (1986), 167–241 [19] N V Quang and L V Thanh, On the strong laws of large numbers for twodimensional arrays of block-wise independent and block-wise orthogonal random variables, Probab Math Statist 25 (2005), 385–391 [20] N V Quang and L V Thanh, On the strong law of large numbers under rearrangements for sequences of blockwise orthogonal random elements in Banach spaces, Aust N Z J Stat 49 (2007), 349–357 [21] A Rosalsky and L V Thanh, Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces, Stoch Anal Appl 24 (2006), no 6, 1097–1117 [22] A Rosalsky and L V Thanh, On almost sure and mean convergence of normed double sums of Banach space valued random elements, Stoch Anal Appl 25 (2007), no 4, 895–911 [23] A Rosalsky and L V Thanh, On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in Rademacher type p Banach spaces, Probab Math Statist 27 (2007), no 2, 205–222 Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 800 N V Quang, L V Thanh and N D Tien [24] A Rosalsky and L V Thanh, Weak laws of large numbers of double sums of independent random elements in Rademacher type p and stable type p Banach spaces, Nonlinear Anal 71 (2009), no 12, 1065–1074 [25] J Rosi´nski, Remarks on Banach spaces of stable type, Probab Math Statist (1980), no 1, 67–71 [26] R T Smythe, Strong laws of large numbers for r-dimensional arrays of random variables, Ann Probability (1973), no 1, 164–170 [27] U Stadtmüller and L V Thanh, On the strong limit theorems for double arrays of block-wise M-dependent random variables, Acta Math Sinica, English Series 27 (2011), no 10, 1923–1934 [28] R L Taylor, Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture Notes in Math 672 Springer, Berlin, 1978 [29] L V Thanh, On the strong law of large numbers for d -dimensional arrays of random variables, Electron Comm Probab 12 (2007), 434–441 [30] N N Vakhania, V I Tarieladze and S A Chobanyan, Probability Distributions on Banach Spaces, Mathematics and its Applications (Soviet Series) 14, D Reidel Publishing Co., Dordrecht, 1987 [31] W A Woyczy´nski, Geometry and martingales in Banach spaces II Independent increments, in: Probability on Banach Spaces, Adv Probab Related Topics 4, Dekker, New York (1978), 267–517 Received May 26, 2010 Author information Nguyen Van Quang, Department of Mathematics, Vinh University, Vinh, Nghe An, Vietnam E-mail: nvquang@hotmail.com Le Van Thanh, Department of Mathematics, Vinh University, Vinh, Nghe An, Vietnam; and Department of Mathematics, National University of Singapore, Singapore E-mail: matlvt@nus.edu.sg Nguyen Duy Tien, Department of Mathematics, Faculty of Natural Sciences, Vietnam National University, Vietnam E-mail: nduytien2006@yahoo.com Brought to you by | University Library Technische Universitaet Muenchen Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM ... maximal inequality for double arrays of p-orthogonal random elements in Banach spaces For a proof see Móricz, Su and Taylor [17] Lemma 2.3 Let ¹Vij W Ä i Ä m; Ä j Ä nº be a double array of random elements. .. Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 789 Almost sure convergence for double arrays in Banach spaces Theorem 3.3 Let ¹Vmn W m 1; n 1º be a double array of random elements in. .. Authenticated | 129.187.254.46 Download Date | 7/27/13 5:54 PM 795 Almost sure convergence for double arrays in Banach spaces Then ¹Vmn W m 1; n 1º is a double array of independent symmetric random elements