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PROBABILITY AND MATHEMATICAL STATISTICS Vol. 27, Fasc, 2 (2007), pp. 205-222 ON THE STRONG LAW OF LARGE NUMBERS FOR SEQUENCES OF BLOCKWISE INDEPENDENT AND BLOCKWISE p-ORTHOGONAL RANDOM ELEMENTS IN RADEMACHER TYPE p BANACH SPACES BY ANDREW ROSALSKY AND LE VAN THANH (GAINESVILLE,FLORIDA) (VINH CITY, VIETNAM) Abstract. For a sequence of random elements {~, n ~ 1} taking values in a real separable Rademacher type p (1 ~ p ~ 2) Banach space and positive constants b; i 00, conditions are provided for the strong law of large numbers 2:;=1 Vi/b" ---+ 0 almost surely. We treat the fol- lowing cases: (i) {V,,, n ~ 1} is blockwise independent with EV" = 0, n ~ 1, and (ii) {~, n ~ 1} is blockwise p-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of M6ricz [12], M6ricz et al. [13], and Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative examples or counterexamples are provided. 2000 AMS Mathematics Secondary: 60Bll, 60B12. Subject Classification: Primary 60F15; Key words and phrases: Blockwise independent random elements, blockwise p-orthogona1 random elements, strong law of large numbers, almost sure convergence, Rademacher type p Banach space. 1. INTRODUCTION M6ricz [12] introduced the concept of blockwise independence for a sequence of (real-valued) random variables and extended a classical strong law of large numbers (SLLN) of Kolmogorov (see, e.g., Chow and Teicher [6], p. 124) to the blockwise independent case. (Technical definitions needed in this paper will be discussed in Section 2.) Gaposhkin [7] and [8] also studied the SLLN problem for sequences of blockwise independent random variables; in those papers he also proved SLLNs for sequences of blockwise orthogonal random variables. In the present paper, we consider a sequence of random elements {~, n ~ 1} defined on a probability space (Q, !F, P) and taking values in a real 206 A. Rosalsky separable SLLN: Banach space lim n.....oo f![ and Le Van Thanh with norm 11'11. We provide conditions I~-i Vi l"b =0 for the almost surely (a.s.), n where {bn, n ~ 1} is a sequence of positive constants with b; i 00. The Banach space f![ is assumed to be of Rademacher type p (1 ~ p ~ 2). The main findings are Theorems 3.1, 3.2, and 3.3. In Theorem 3.1 the random elements {li;" n ~ 1} are assumed to be blockwise independent with Eli;, = 0, n ~ 1, whereas in Theorem 3.3 the random elements are assumed to be blockwise p-orthogonal. In Theorem 3.2, it is shown that the implication (3.2) ~ (3.3) in Theorem 3.1 indeed provides an exact characterization of Rademacher type p Banach spaces. Theorems 3.1 and 3.3 are very general results in that they are new when the Banach space f![ is the real line R. Of course, special cases of Theorems 3.1 and 3.3 are known to hold when f![ = R. The present work extends results of M6ricz [12], M6ricz et al. [13], and Gaposhkin [8]. Our proofs are substantially different from those of the earlier counterparts due to a recent and elementary result of Chobanyan et al. [5] (Lemma 2.5 below) and these differences will be discussed in Remark 3.2. The plan of the paper is as follows. Technical definitions, notation, lemmas and other results used in the proofs of the main results or their corollaries are given in Section 2. The main results are stated and proved in Section 3. In Section 4, some corollaries and interesting examples or counterexamples are presented. 2. PRELIMINARIES Some definitions, notation, and preliminary results will be presented prior to establishing the main results. Let f![ be a real separable Banach space with norm 11'11. A random element in f![ will be denoted by V or li;" etc. The expected value or mean of a random element denoted by EV: is defined to be the Pettis integral provided it exists. That is, V has an expected value EV Ef![ if f(EV) = E(f(V)) for every f Ef![*, where f![* denotes the (dua0 space of all continuous linear functionals on f![. If E IIVII < 00, then (see, e.g., Taylor [18], p. 40) V has an expected value. But the expected value can exist when E IIVII = 00. For an example, see Taylor [18], p. 41. Let {~, n ~ 1} be a symmetric Bernoulli sequence; that is, {~, n ~ 1} is a sequence of independent and identically distributed (i.i.d.) random variables with P {Yi = 1} = P {Yi = -1} = 1/2. Let f![oo = f![ x f![ x f![ x .. , and define v: 00 ~ (f![) = {(Vi, V2, ... ) E f![00: I~ Vn n= 1 converges in probability}. 207 Strong law of large numbers in Banach spaces Let 1 ~ p ~ 2. Then q; is said to be of Rademacher a constant 0 < C < 00 such that OC! E II I type p if there exists OC! P Y;, vnll C ~ n=1 I Ilvnll P for all (V1' ... ) E ~ V2, (q;). n=1 Hoffmann-Jorgensen and Pisier [9] proved for 1 ~ p ~ 2 that a real separable Banach space is of Rademacher type p if and only if there exists a constant o < C < 00 such that n (2.1) Ell n I ViII P I EIIViW C ~ i=1 i=1 for every finite collection {V1, ••• , ~} of independent mean 0 random elements. If a real separable Banach space is of Rademacher type p for some 1 < p ~ 2, then it is of Rademacher type q for all 1 ~ q < p. Every real separable Banach space is of Rademacher type (at least) 1 while the .Pp-spaces and lp-spaces are of Rademacher type 2 /\ P for p ;?: 1. Every real separable Hilbert space and real separable finite-dimensional Banach space is of Rademacher type 2. In particular, the real line R is of Rademacher type 2. A finite collection of random elements {V1, ... , VN} (N ;?: 2) is said to be p-orthogonal (1 ~ p < (0) if E II~IIP < 00 for all 1 ~ n ~ Nand m n E II I i=1 an(i) ~(i)IIP ~ E II I P an(i) Vn(i)II i=1 for all choices of 1 ~ n < m ~ N, for all constants {af, ... , am}, and for all permutations TC of the integers {1, ... , m}. A sequence of random elements {~, n > 1} is said to be p-orthogonal (1 ~ p < (0) if {V1, ... , VN} is p-orthogonal for all N ;?: 2. The notion of p-orthogonality was introduced by Howell and Taylor [10]; we refer to Howell and Taylor [10] and M6ricz et al. [13] for a detailed discussion of p-orthogonality. Let {13k, k ;?: 1} be a strictly increasing sequence of positive integers with 131 = 1 and set Bk = [13k, 13k + 1), k;?: 1. A sequence of random elements {~, n ;?: 1} is said to be blockwise independent (resp., blockwise p-orthogonal (1 ~ p < (0)) with respect to the blocks {Bk, k ;?: 1} if for each k ;?: 1 the random elements {Vi, i E Bk} are independent (resp., p-orthogonal). Thus the random elements with indices in each block are independent (resp., p-orthogonal) but there are no independence (resp., p-orthogonality) requirements between the random elements with indices in different blocks; even repetitions are permitted. The following notation will be used throughout this paper. For x ;?: 0, let Lx J denote the greatest integer less than or equal to x and let Ix l denote the smallest integer greater than or equal to x. We use Log to denote the logarithm to base 2. The symbol C denotes a generic constant (0 < C < (0) which is not necessarily the same in each appearance. 208 A. Rosalsky For {13k, k ;::: 1} and {Bb notation: Bkm) = B and Le Van Thanh k ;::: 1} as above, we introduce n [Z", 2m + 1), k k ;:::1, m ;:::0, = {k;::: 1: Bkm) # 0}, L; rkm) = min Bkm) , m;::: 0, m kEIm, = card 1m, Cm the following > 0, m;::: 0, 00 q>(n) I = Cm I[2m,2m+ (n), 1) n ;:::1, m=O denotes the indicator function of the set [Z", 2m+i), m It is easy to verify that the following relations are satisfied: (i) If 13k = 2k- \ k;::: 1, then where I[2m,2m+1) (2.2) Cm (ii) If 13k = L = 1, m > 0, «: (2.3) J Cm (iii) If 13k (2.4) (iv) If 13k (2.5) (1) = (!) L 2k/(Logk)C< J Cm = (D((Logm)a) and q>(n) and = (D(m(i-a)/a) Lka J , = Cm = = 1, n > 1. k = (!) (1). for all large k, where ex > 0, then q>(n) = (D((Log Log nY). = L 2kC<J for all large k, where (2.6) (vi) If 13k q>(n) for all large k, where q > 1, then = Cm (v) If 13k and and q>(n) °< ex < 1, then = (D((Log n)(i-a)/a). > 1, where ex> 1, then = (D(2m/a) and q>(n) = (D(ni/a). k, k;::: 1, then (2.7) Cm = 2m, m > 0, and q>(n) ~ n, n The following result is well known when > 1. = Rand f!{ j (2.8) E(( max II I Villt) ~ C i~j~n where p = 2. ° 2.1. Let {~, n ;:::1} be a sequence of independent mean random in a real separable Rademacher type p (1 ~ p ~ 2) Banach space. Then LEMMA elements > 0. the constant i=i C does not depend n I EIIViII i=i P , n > 1, on n. Proof. Let!F" = o-(Xb 1 ~ i ~ n), n ;::: 1. Now it is well known but seems to have been first observed by Scalora [17] that 1 Viii, !F", n ;::: 1} is a real submartingale. In the case 1 < p ~ 2, by Doob's submartingale maximal {III;= 209 Strong law of large numbers in Banach spaces inequality (see, e.g., Chow and Teicher [6J, p. 255), for n ~ 1 we have n ~ c I E IIViII P (by (2.1)), i=i establishing (2.8). In the case p = 1, note that for n ~ 1 j E( max i~j~n j IIi=iI ViiI) ~ again establishing (2.8). E( max i~j~ni=i n n I IIViiI) = E( i=iI IIViIl) = i=iI E IIViIl, II PROPOSITION 2.1 (Hoffmann-Jorgensen and Pisier [9]). Let 1 ~ p ~ 2 and let PI be a real separable Banach space. Then the following two statements are equivalent: (i) PI is of Rademacher type p. (ii) For every sequence {~, n ~ 1} of independent mean 0 random elements in PI, the condition implies ,,~ V: L.J=l J n -+ 0 a.s. LEMMA2.2 (Howell and Taylor [10J). If PI is a real separable Rademacher type p (1 ~ p ~ 2) Banach space, then there exists a constant C < 00 such that n E n III ViII P I C ~ i=i E IIVi liP , n ~ 1, i=i for all p-orthoqonal sequences of PI-valued random elements. LEMMA2.3 (Moricz et al. [13J). Let {~, n ~ 1} be a p-orthogonal (1 ~ p < 00) sequence of random elements in a real separable Banach space and suppose that there exists a sequence of nonnegative numbers {un, n ~ 1} such that m m EIIIViII P i=n ~ I Ui for all m ~ n ~ 1. i=n Then P m j E(( m~x n~J~m IIi=n I Villt) ~ (Log(2(m-n+ 1))) I Ui, m ~ n ~ 1. i=n An immediate consequence of Lemmas 2.2 and 2.3 is that if {Vi, ... , VN} is a p-orthogonal (1 ~ p ~ 2) collection of N ~ 2 random elements in a real 210 A. Rosalsky separable Rademacher type p Banach space, then and Le Van Thanh j (2.9) E (m~x n~J~m III ViIIY i=n P 1))) ~ C(Log(2(m-n+ m I E IIViII P, 1~ n~ m ~ N, i=n where the constant C does not depend on n, m, or N. The next lemma is most probably known but we are unable to locate a reference. It follows immediately from HOlder's inequality writing ai = ai'l, as was so kindly pointed out to us by the referee. LEMMA2.4. If a, ~ 0, 1 ~ i ~ n, and p ~ 1, then n n (I aiY ~ I nP-1 i= 1 i= af· 1 The following elementary result was recently obtained by Chobanyan et al. [5] and it will playa key role in the proofs of Theorems 3.1 and 3.3. Lemma 2.5 is closely related to a classical result of Prokhorov [14] and [15] when f!( = R, the {~, n ~ I} are independent symmetric random variables, and b; = n", n ~ 1, with IX > O. LEMMA2.5 (Chobanyan et al. [5]). Let {~, n ~ I} be a sequence of random elements in a real separable Banach space f!{, let {bn, n ~ I} be a nondecreasing sequence of positive constants, and let {kn, n ~ O} be a sequence of positive integers such that and (2.10) Then · 1rm (2.11) n--+oo if and only (2.12) "'~ .L..t= b 1 ll;t n = 0 a.s. if . maxkn~k<kn+lIII:=kn 1im b b n --+ 00 kn + 1 - Vill_ - 0 a.s. kn Remark 2.1. (i) Note that the first inequality of (2.10) ensures that {kn, n ~ O} is strictly increasing and limn--+oo b; = 00. (ii) It follows that if (2.12) holds for some sequence of positive integers {kn, n ~ O} satisfying (2.10), then (2.12) holds for every sequence of positive integers {kn, n ~ O} satisfying (2.10). Thus, in order to prove the SLLN (2.11), nothing is lost in working with a convenient sequence such as k; = 2n, n ~ O. This remark was made by Chobanyan et al. [5]. Strong law of large numbers in Banach spaces 211 (iii) If the random elements {lI;" n ~ 1} are independent and symmetric, then by the random element version of Levy's inequality (see, e.g., Araujo and Gine [3], p. 102) and the Borel-Cantelli lemma, (2.12) is equivalent to the structurally simpler and apparently weaker condition "kn+ · II '--'i=kn 1tm b 1 b kn+ 1 - n-r co 17;11 -1 l - - ° a.s. kn 3. THE MAIN RESULTS With the preliminaries accounted for, the main results may now be established. When f£ = Rand b; == n, a version of Theorem 3.1 was obtained by Gaposhkin [8] using a substantially more complicated argument. Gaposhkin's [8] condition is slightly different from (3.2) specialized to f£ = Rand b; == n and is at least as strong. Gaposhkin's [8] result is an extension of an earlier result of M6ricz [12] which was apparently the first SLLN for a sequence of blockwise independent random variables. ° THEOREM 3.1. Let {lI;" n ~ 1} be a sequence of mean random elements in a real separable Rademacher type p (1 ~ p ~ 2) Banach space f£ and let {bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1) and If {lI;" n ~ and if 1} is blockwise independent with respect to the blocks {Bk, k ~ 1} (3.2) then (3.3) lim n-r co Proof. I~-l Vi lb = ° a.s. n Set j 1k(m) = max II jeB~m) I Viii, kEIm' m ~ 0, i = r~m) and Note that, for m ~ 0, ET! ~ -bP1 m 2 + 1 C~-l I kelm E(1k(m»)p (by Lemma 2.4) 212 A. Rosalsky ~ ~C~-l 2m+ 1 C and Le Van Thanh I I EIIViII P ua.; (by Lemma 2.1) ieBkm) I:=o It thus follows from (3.2) that ETl: < ty and the Borel-Cantelli lemma lim Tm 00, and so by the Markov inequali- = 0 a.s. m--+w Now it follows from the first inequality of (3.1) that max2m~k<2m+lIII:=2m Viii ~ Cmax2m~k<2m+lIII:=2m --------- b2m+l-b2m The conclusion Viii b2m+l (3.3) follows immediately ~ CTm from Lemma 2.5. ~ 0 a.s. III Remark 3.1. (i) The slower b; i 00, the stronger is the assumption (3.2), but so is the conclusion (3.3). (ii) Theorem 3.1 is an analogue of an SLLN of Adler et al. [2] obtained for a sequence of independent random elements in a Rademacher type p (1 ~ p ~ 2) Banach space. The Adler et al. [2] result, which extends the implication (i) => (ii) in Proposition 2.1 to more general norming constants o < b; i 00, is a random element analogue of a classical result of Kolmogorov (see, e.g., Loeve [11], p. 250). It should be pointed out that this SLLN of Adler et al. [2] does indeed follow immediately from Theorem V.7.S (or Corollary V.7.S) of Woyczynski [19] and the Kronecker lemma. (iii) When p = 1, Theorem 3.1 is not of interest since the mean 0 assumption, the blockwise independence assumption, and (3.1) are not needed. Indeed, for a sequence of random elements {~, n ~ 1} in a real separable Banach space and constants 0 < b; i 00, Cantrell and Rosalsky [4] recently proved that if IE ( llViII+b IIViii ) < assumption than I: E Vill/b 00 i=l (which is a weaker i 1 II i < (0), then n lim n--+w I Vi/b n = 0 a.s. i= 1 irrespective of the joint distributions of the {~, n ~ (iv) If (3.1) holds where 0 < b, i and (3.4) Cm = o (am) for all a> 1, 1}. Strong law of large numbers then a sufficient condition 213 spaces for (3.2) to hold is that EllliiliP 00 (3.5) in Banach i~~<oo for some O<q<p. To see this, it follows from the first inequality of (3.1) that b2n+ 1/b2n for some b > 0 and all n ~ O. Thus for all m ~ 1 we have m-1b (3.6) = b, b2m TI -b-. 2j+1 j=O ~ 1+b m ~ bd1 +b) . 2J Hence m ~ m~o i~m ~c I 00 m=O = c 2m+ 1-1 E Ill/;IIP I __ ql_ IE ";"b (by (3.6)) (by (3.4) with a = (1 + b)(P-q)/(p-l») bi i=2m P i=l thereby establishing E IlliillP C~-l b1 bf-q (1 + b)(p-q)m 1 2 + -l 00 < 00 (by (3.5)), i (3.2). In the next theorem, we will show that the implication (3.2) => (3.3) holding indeed completely characterizes Rademacher type p Banach spaces. THEOREM 3.2. Let!!l' be a real separable Banach space and let 1 ~ p ~ 2. Then the following two statements are equivalent: (i) !!l' is of Rademacher type p. (ii) For every sequence of mean 0 random elements {~, n ~ 1}, which is blockwise independent with respect to some sequence of blocks {Bk, k ~ 1} and every nondecreasing sequence of positive constants {bn, n ~ 1} satisfying (3.1), the implication (3.2) => (3.3) holds. Proof. The implication (i) => (ii) is precisely Theorem 3.1. To verify the implication (ii) => (i), assume that (ii) holds. Let {~, n ~ 1} be a sequence of independent mean 0 random elements in !!l' such that (3.7) 214 A. Rosalsky In view of Proposition and Le Van Thanh 2.1, it suffices to verify that I;=1 Vi ~ (3.8) n ° a.s. Note that {~, n ~ 1} is blockwise independent with respect to the blocks {[2k-1, 2k), k ~ 1}. Set b; = n, n ~ 1. Then (3.1) holds and, recalling (2.2), we see that (3.7) and (3.2) are the same. Thus, by (ii), we see that (3.8) holds. l1li b; Bk The next theorem, when specialized to {~, n ~ 1} being p-orthogonal and n", n ~ 1, where a > 0, reduces to a result of M6ricz et al. [13] by taking = [2k-1, 2k), k ~ 1, and recalling (2.2). = THEOREM 3.3. Let {~, n ~ 1} be a sequence of random elements in a real separable Rademacher type p (1 ~ p ~ 2) Banach space and let {bn, n ~ 1} be a nondecreasing sequence of positive constants satisfying (3.1). If {~, n ~ 1} is blockwise p-orthoqonal with respect to the blocks {Bk, k ~ 1} and if W (cp (i)r , < 00, ,~, E I~"P (Log (3.9) then (3.10) lim ,," LJ'-1 Vi l"b n->CXJ = " ° a.s. Proof. Define 1k(rn), kEIrn' m ~ 0, and Trn, m ~ 0, as in the proof of Theorem 3.1. Note that, for m ~ 0, ETJ ~ -bP1 m C~-1 2 + 1 C ~ ~c~-1 1 I (Log(2cardBkrn»)t C~-1 m 2 + 1-1 i~m I EIlViIIP (by (2.9)) ieBkm) 2m+1-1 I EIIViIlP i=2m 1 ~ C (by Lemma 2.4) C(Log2rn+1)P ~ _p-C~-1 b2m+l b~m+1 E(1k(rn»)p »a ; 2m+ 1 ~ I kelm rn C (Log 2 )P 2m+1-1 i~m EIlViIIP E 11T7;IIP 1 T(Logi)P(CfJ(i)t- . I:=o It thus follows from (3.9) that ETJ < 00. The rest of the argument exactly the same as that at the end of the proof of Theorem 3.1. l1li is Remark 3.2. We close this section with discussion of the main difference between the structure of the proofs of Theorems 3.1 and 3.3 and that of the 215 Strong law of large numbers in Banach spaces earlier results by Moricz [12], Moricz et al. [13], and Gaposhkin [8]. As in Moricz [12] and Gaposhkin [8], let us now take b; = n, n ~ 1, and !!l' = R (although Moricz et al. [13] took b; = n", n ~ 1, where (X > 0, and they assumed that the Banach space !!l' is of Rademacher type p (1 ~ p ~ 2». Moricz [12], Moricz et al. [13], and Gaposhkin [8] bounded 1 Vi/n (where 2m ~ n < 2m + 1) by I;= (3.11) I;=l Vii ~ II:~;l2m I Vii n 1 +2m max 2m<k<2m+1 Ii=2I Vii· k m They then argued that each of the two terms on the right-hand side of (3.11) converges to 0 a.s. as m ~ 00. Their arguments that the first term on the right-hand side of (3.11) converges to 0 a.s. as m ~ 00 did not use any (type of) independence hypothesis; only (a type of) orthogonality was used. However, in our proof of Theorems 3.1 and 3.3, we only need to argue that the second term on the right-hand side of (3.11) converges to 0 a.s. as m ~ 00. This is the case because we then apply Lemma 2.5, and so we are in effect bounding 1 Vi/n (where 2m ~ n < 2m+1) by I;= I;=1 Vii ~ f I r=O max2r<k<2r+lII:=2r 2m ~ m n _ - 2m r~o Vii Vii 2r max2r<k<2r+lII:=2r 2r which via the Toeplitz lemma converges to 0 a.s. as m ~ and Theorem 9.1 of Chobanyan et al. [5]). 4. COROLLARIES ' 00 (see Proposition 3.1 AND EXAMPLES In this section, some particular cases of Theorems 3.1 and 3.3 are presented as corollaries. Some illustrative examples or counterexamples are also provided. When the underlying Banach space is the real line, p = 2, and b; == n, Corollaries 4.1, 4.3, 4.4 (take (X2 = 1), and 4.5 reduce to results of Gaposhkin [8]. This special case of Corollary 4.1 is also the SLLN for a sequence of blockwise independent random variables obtained by Moricz [12]. In view of Remark 3.1 (iii), we formulate Corollaries 4.1-4.5 assuming that p > 1. COROLLARY 4.1. Let {It;,, n ~ 1} be a sequence of mean 0 random elements in a real separable Rademacher type p (1 < p ~ 2) Banach space and let {bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1) holds. If {It;,, n ~ 1} is blockwise independent with respect to the blocks {[2k-1, 2k), k ~ 1} (or, more generally, with respect to the blocks {[,Bk' ,Bk+ 1), 216 A. Rosalsky k ~ 1}, where 13k= L qk-l J and Le Van Thanh for all large k. and q > 1) and co (4.1) if E IIViIIP i~l~<oo, then (3.3) holds. Proof. Recalling (2.2) and (2.3), we infer that the assumption (4.1) ensures that (3.2) holds. The conclusion follows directly from Theorem 3.1. l1li COROLLARY 4.2. Let {~, n ~ 1} be a sequence of mean 0 random elements in a real separable Rademacher type p (1 < p ~ 2) Banach space and let {bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1) holds. If {~, n ~ 1} is blockwise independent with respect to the blocks {[13k, 13k+ 1), k ~ 1}, where 13k= L 2k/(Logk)'" J for all large k and a > 0, and if ~ EIIViIIP(L (4.2) i~2 ~ L og ')IX(p-l) og z < 00, then (3.3) holds. Proof. Recalling (2.4), we see that the assumption (4.2) ensures that (3.2) holds. The conclusion follows directly from Theorem 3.1. l1li COROLLARY 4.3. Let {~, n ~ 1} be a sequence of mean 0 random elements in a real separable Rademacher type p (1 < p ~ 2) Banach space and let {bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1) holds. If {~, n ~ 1} is blockwise independent with respect to the blocks {[13k, 13k+ 1), k ~ 1}, where 13k= L 2k'"J for all large k: and 0 < a < 1, and if IEll ;II P (4.3) i=l (Log 1 i)(1X- -1)(p-1) < 00, bi then (3.3) holds. Proof. By (2.5) the assumption (4.3) ensures that (3.2) holds. The conclusion follows directly from Theorem 3.1. l1li 4.4. Let {~, n ~ 1} be a sequence of mean 0 random elements in a real separable Rademacher type p (1 < p ~ 2) Banach space and suppose that {~, n ~ 1} is blockwise independent with respect to the blocks {l Lk J, L(k+ 1) J), k ~ 1}, where al > 1. Let a2 ~ ail. If COROLLARY IX1 1X 1 IE ".;" P (4.4) i= 1 where q = (p (al a2 -1) + l)/al' < 00, t then 1· I;=ln Hfl n~oo Vi IX 2 = 0 a.s. Strong Proof. law of large numbers in Banach spaces 217 Recalling (2.6), we obtain P ~ E II ViII ( (i))P-1 ~ C ~ E IIVillP i(p-1)/cq L. 'a2P tp "'" L. 'a2P i=1 1 i=1 1 = C IE ".;IIP < and the conclusion Remark that !X2 :::; q 00 (by (4.4)) 1 i= 1 follows from Theorem 3.1. III 4.1. Apropos of the constant q in Corollary 4.4, it may be noted < !X2 p. Counterparts to Corollaries 4.1-4.4 can easily be given when {~, n ;:::1} is blockwise p-orthogonal with respect to a sequence of blocks. We content ourselves with only presenting the counterpart to Corollary 4.1. Corollary 4.5 is an extension of the classical SLLN resulting from the celebrated RademacherMensov fundamental convergence theorem for sums of orthogonal random variables (see, e.g., Revesz [16], pp. 86-87). COROLLARY 4.5. Let {~, n ~ 1} be a sequence of mean 0 random elements in a real separable Rademacher type p (1 < p :::;2) Banach space and let {bn, n ~ 1} be a nondecreasing sequence of positive constants such that (3.1) holds. If {~, n ~ 1}is blockwise p-orthoqonal with respect to the blocks k 1 k {[2 - , 2 ), k ~ 1} (or, more generally, with respect to the blocks {[13k, 13k + 1), k ~ 1}, where 13k = L q'c-1 J for all large k and q > 1) and if E IIViIIP i~1 bf(Logl) 00 (4.5) .P < 00, then (3.10) holds. Proof. Recalling (2.2) and (2.3), we infer that the assumption (4.5) ensures that (3.9) holds. The conclusion follows directly from Theorem 3.3. III We close by presenting four examples. The first example illustrates Theorem 3.1 and Corollary 4.1. EXAMPLE 4.1. Let {~, n ~ 1} be a sequence of independent mean 0 random elements in a real separable Rademacher type p (1 :::;p :::;2) Banach space and suppose that I i=1 00 (4.6) EllmW < 'ap 1 00 2m:::; n < 2m+1, {~, n ~ 1} is blockwise independent Let 5 - ~ = PAMS 27.2 ~-2m+1' for some !X > O. m ~ O. Then E~ = 0, n > 1, and with respect to the blocks {Ctc-1, 2k), 218 A. Rosalsky and Le Van Thanh k ~ 1}. Now, using (4.6), we obtain ~ E IIViIIP = ~ 2m~ 1..J '(J.p 1..J 1..J i=l 1 m=O i=2m 1 11U'i-2 +lIIP E m '(J.p 1 3.1 (or by Corollary 4.1), Thus, by (2.2) and Theorem ,,~ l7; · 1..Jl=l(J. l 1Hfl n-> = n 00 Finally, we note that, by Theorem ,,~ a.s. 1 of Adler et al. [2J, we also have JIJf; 1..Jl=n(J. 1 l li n~ ° = ° a.s. The second example shows that Theorem 3.1 can fail if the series in (3.2) diverges. More specifically, Example 4.2 shows that we cannot replace (3.2) by the weaker condition EII~IIP(cp(n)t-l/b~ = 0(1). ° EXAMPLE4.2. Let {~, n ~ 1} be a sequence of independent mean random elements in a real separable Rademacher type p (1 ::::;; p ::::;;2) Banach space where P{II~II 1 = 2n-1} = 1+Logn ' Let lI;, = ~-2m+l' 2m ::::;;n < 2m+\ m n ~ 1, and {~, n ~ 1} is blockwise {[2k-1, 2k), k ~ 1}. Now V2m+l-1 = a sequence of independent random P{~ 1 1 +Logn' = O} = 1 n ~ 1. ~ 0, and let bn = n, n ~ 1. Then Ell;, = 0, independent with respect to the blocks W2m, m ~ 0, and so {V2m+l-1, m ~ o} is elements with P {V2m+l-1 = o} m = --1' m+ m~O. Then, recalling (2.2), we obtain ~ EIIViIIP( .1..J l=l b-P l (.))P-l cp 1 ~ s-: ~ EIIV2m+1-11IP 1..J (2m + 1-l)P m=O 00 (2m + 1_ l)P = m~o(m+1)(2m+l-1)p 00 = m~om+1 1 = 00, 219 Strong law of large numbers in Banach spaces and so (3.2) fails. Moreover, by the Borel-Cantelli since lemma we have Thus, 1 -1' - Imsup 2m+1-111 ~ u IIVm+l II~II 1 -...;;:: Imsup 2 m-+co ____ l' ~ nn sup 11-+ co - n n-+co III;= ~II+ uim 1 n III;~:1~II a.s. sup n- n-+co implying III;=b ~II_- l'im sup III;= · sup 1im 1 n-+co 1 ~" n n-r n co -:/ -1 a.s. 2 '- Thus (3.3) fails. We also note that for all large n, writing 2m again recalling (2.2), we get E II~IIP( bP n ((J ~ n < 2m+1 and = E 11~-2m+ l11P ())P-l n w (2(n-2m+ 1)-1t nP (1 + Log(n-2m+ 1)) + (2(2m+1-1-2m 1)-1t ~--'------------'--mp m 1 m 2 (1 Log(2 + -1-2 1)) + ~ -- 2P m+1 + as n -+ 0 -+ CIJ. The third example shows that the Rademacher type p hypothesis in Theorem 3.1 cannot in general be dispensed with. Example 4.3 concerns the real separable Banach space 11 of absolutely summable real sequences v = {Vj,j ~ 1} with norm Ilvll= I:1IvJ The element of 11having 1 in its nth position and 0 elsewhere will bel denoted by v(n), n ~ 1. EXAMPLE4.3. Consider the real separable Banach space 11. It is well known (see, e.g., Adler et al. [1]) that 11 is not of Rademacher type p for any 1 < p ~ 2. Define a sequence of random elements {~, n ~ 1} in 11 by requiring the {~, n ~ 1} to be independent with p {~ = v(n)} = p {~ = - v(Il)} = 1, n ~ 1. 220 A. Rosalsky and Le Van Thanh Let ~ = ~ _2m + 1, 2m ~ n < 2m + 1 , m ~ 0. Let 1 < p ~ 2 and let b; = nIX,n ~ 1, where < a ~ 1. Then E~ = 0, n ~ 1, and {~, n ~ 1} is blockwise independent with respect to the blocks {[2k-1, 2k), k ~ 1}. Now, by (2.2), «: IEII;IIP (cp(oy-1 = I ':P < b i=1 00 i=11 i since ap > 1, and so (3.2) holds. Next, for MEN, +1- 1 +1- 1 +1- 1 Ii=1 Vi=I m=O i=2I Vi=I m=O i=2I 2M M 2m M 2m m M 2M I I = m 2m Wii = m=O i=1 2M M I I I Wii = i=1 m=iLogil 1) Wii, (M -iLogil+ i=1 and so III~~1+1-1 Viii III~~1(M -iLogil+ 2M = Consequently, 1) Wii'l (2M+1_1)IX b 2M+1-1 Ii=1 (M -iLogil+ 1).... 2M ~ 2M+1 (2M+1_1)IX 1 ="2 a.s. (3.3) fails. The final example, which is a modification of an example of M6ricz [12J, shows apropos of Corollary 4.1 that under its hypotheses the series 1 Vi/bi can diverge a.s. Consequently, the conclusion of Corollary 4.1 (or Theorem 3.1) cannot in general be reached through the well-known Kronecker lemma approach for proving SLLNs. I;= 4.4. Let the underlying Banach space be the real line and let p = 2. Let {Xn, n ~ 1} be a sequence of independent mean random variables such that P {X 1 =I- O} = 1 and 1 EX~ < 00. Define for n ~ 1 ° EXAMPLE I:= v: _ n - {nXt!(l nXn, +Logn), LognEN, Logn¢N, and let b; = n, n ~ 1. Then E~ = 0, n ~ 1, and {~, n ~ 1} is blockwise independent with respect to the blocks {[2k-1, 2k), k ~ 1}. Now Ev:2 1 I -+ = EXt m=o(1+m) I n=1 bn C1) C1) C1) 2 + I n=3 EX~ < 00, Logn¢N and so, by Corollary 4.1, (3.3) holds. 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Taylor, Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture Notes in Math. No 672, Springer, Berlin 1978. [19] W. A. Wo ycz yri sk i, Geometry and martingales in Banach spaces. Part II: Independent increments, in: Probability on Banach Spaces, 1. Kuelbs (Ed.), Advances in Probability and Related Topics, P. Ney (Ed.), Vol. 4, Marcel Dekker, New York 1978, pp. 267-517. Department of Mathematics Vinh University Vinh City, Nghe An Province Vietnam E-mail: [email protected] Department of Statistics University of Florida Gainesville, Florida 32611, USA E-mail: [email protected] Received on 7.10.2005; revised version on 31.5.2006