Best Possible Sufficient Conditions for Strong Law of Large Numbers for Multi-Indexed Orthogonal Random Elements

It will be shown and induced that the d-dimensional indices in the Banach spaces version conditions ∑ n(E‖Xn‖/|n|) < ∞ are sufficient to yield lim min 1≤ j≤d(nj)→∞(1/ |nα|)∑k≤n ∏d j=1(1− (kj − 1)/nj)Xk = 0 a.s. for arrays of James-type orthogonal random elements. Particularly, it will be shown also that there are the best possible sufficient conditions for multi-indexed independent real-valued random variables.


Introduction
The laws of large numbers for orthogonal random variables or Banach space-valued random elements are investigated by several authors.A consequence of Rademacher-Menshov theorem [2,3] showed that the sufficient condition for a strong law of large numbers of a sequence of orthogonal real-valued random variables with 0 means and finite second moments is ∞ k=1 (σ 2 k /k 2 ) • [log 2 (k + 1)] 2 < ∞. Warren and Howell [4] proposed the sufficient condition ∞ k=1 (E X k 1+α /k 1+α ) • log 1+α k < ∞, 0 < ∞ ≤ 1, for strong convergence of the one-dimensional B-valued James-type orthogonal random variables.M óricz [5] showed that ∞ i=1 ∞ k=1 (σ 2  ik /i 2 k 2 ) • [log 2 (i + 1)] 2 [log 2 (k + 1)] 2 < ∞ is the necessary condition for the strong convergence for arrays of quasi-orthogonal real-valued random variables.M óricz [6] obtained a sufficient condition for strong limit theorems for arrays of quasi-orthogonal real-valued random variables.M óricz et al. [7] showed that the sufficient condition for the strong convergence of (1/m α n β ) m i=1 n k=1 X ik for arrays of orthogonal, type p Banach space-valued random elements is p log 2 (k + 1) p < ∞. (1.1)In this paper, the strong laws of large numbers will be investigated for James type of orthogonality in a Banach space.In order to induce d-dimensional case, d > 2, we wish to investigate the strongly convergent behavior of a more general Cesaro-type means, (1/m α n β ) m i=1 n k=1 (1 − (i − 1)/m)(1 − (k − 1)/n)X ik , as m,n → ∞, for arrays of twodimensionally indexed orthogonal random elements in a Banach space of type p, 1 ≤ p ≤ 2, and 1/2 < α, β ≤ 1, though Su [1] showed a case of α = 1 = β.In particular, it will be proven that the sufficient conditions are also the best possible even for independent real-valued random variables.The definition for an array of orthogonal random elements and the formulation of previous results and auxiliary lemmas for orthogonality are given in Section 2. The major results and their proofs are in Sections 3 and 4, respectively.

Preliminaries and auxiliary lemmas
The basic definitions and properties of Banach space-valued random variables (or random elements) are well established in the literature (e.g., [8]).In these preliminaries, we only introduce the concepts which are necessary and not easy to read in the literature.
Our sense of orthogonality throughout this manuscript is that of James type orthogonality.For elements x and y in a Banach space B, x is said to be James orthogonal to y (denoted by x ⊥ J y) if x ≤ x + ty for all t ∈ .If B is a Hilbert space, then James type orthogonality agrees with the usual notion of orthogonality where the inner product is 0 since x + ty 2 = (x + ty,x + ty) = x 2 + t 2 y 2 + 2t(x, y) ≥ x 2 for all t ∈ if and only if (x, y) = 0.However, in a Banach space where the norm is not generated by an inner product, it is possible for x ⊥ J y but y ⊥ J x with (x, y) = 0.For instance, let 2 = {(x 1 ,x 2 ) : (x 1 ,x 2 ) = |x 1 | + |x 2 |, x 1 ,x 2 ∈ } and x = (2,0) and y = (2,−2).Then, it is clear that the usual inner product (x, y) = 4 = 0. Next, x ⊥ J y but y ⊥ J x since x + ty = |2 + 2t| + | − 2t| ≥ 2 = x and y + tx = |2 + 2t| + | − 2| = 3 < y = 4 while picking t = −1/2.Therefore, it is not possible to create a notation of orthogonality with a good geometrical meaning in an arbitrary Banach space without the inner product.As a result, James-type orthogonality is adopted to circumvent this shortcoming [7].
Let {X ik , i,k ≥ 1} be a double sequence of random elements in the Banach space L p (B) with zero means, that is, E(X ik ) = 0 for all i, k and finite pth moments, E X ik p < ∞ for all i, k, where • is the norm of the separable Banach space B. The following is the extended definition for arrays of orthogonal random variables in Banach spaces.
In retrospect, a separable Banach space B is of type p (1 ≤ p ≤ 2) if and only if there is a constant C > 0 (depending on B only) such that E n i=1 X i p ≤ C n i=1 E X i p when {X i } are independent random elements with zero means and finite pth moments [8].In Kuo-Liang Su 3 order to obtain the desired results, a useful version of moment inequality for arrays that extend the results of Howell and Taylor [9]  (2.5) We also need two more crucial lemmas as follows; they are extended from Kronecker's lemma and Shiryayev [10], respectively, and will be proven in Section 4.

Lemma 2.5 (two-dimensionally indexed version of Kronecker's lemma).
Let {a m } and {b n } be sequences of positive increasing numbers, both a m ↑ ∞ and b n ↑ ∞ when m → ∞ and n → ∞, respectively.Let {x i j ; i, j ≥ 1} be an array of positive numbers such that (2.6) Lemma 2.6.A sufficient and necessary condition that ζ mn → 0 with probability one as m,n → ∞ is that for any ε > 0, (2.7)

Major results
Theorems 3.1 and 3.2 are two-dimensionally indexed versions of strong convergence for Cesaro-type means for arrays of Banach space-valued random elements and hence their proofs are more complicated than that in real cases because of the structure of spaces.
Theorem 3.1.Let {X ik } be an array of orthogonal (in L p (B)) random elements with zero means in a Banach space B of type p, for some Theorem 3.2.Let {X ik } be an array of orthogonal (in L p (B)) random elements with zero means in a Banach space B of type p, for some The generalization to d-dimensional arrays random elements of the previous two theorems can be obtained easily by the same methods [1].Theorems 3.3 and 3.4 are to show that the sufficient conditions in the previous theorems are the best possible conditions for independent real-valued random variables, since the real line is of type p, 1 ≤ p ≤ 2.

Proofs
Here we will verify Lemmas 2.5 and 2.6 first, then prove the case of d = 2, that is, Theorems 3.1, 3.2, 3.3, and 3.4.We may apply the analogous approaches for the d-dimensional cases, d > 2.
Proof of Lemma 2.5.
Then by the one-dimensional version Kronecker's lemma, we can conclude that (1/a m ) m i=1 (x i j /b j ) → 0 as m → ∞, for every j, and (1/b n ) n j=1 (x i j /a i ) → 0 as n → ∞, for every i.Hence, for any Similarly, for any (ii) Apparently, for any ε > 0, when m > M (say), we can conclude that Proof of Lemma 2.6.Fix any m ≥ 1, for any ε > 0, let A ε mn = {ω : sup i≥m ζ in > ε} and However, P(A ε m ) = lim n P( k≥n A ε mk ).Hence, for any fixed m ≥ 1, Proof of Theorem 3.1.We need some useful arguments in the proof in [1,7], and Lemmas 2.3 and 2.4.For positive integers u and v, for any ε, where We have where A (1)  rs = max 2 r <m≤2 r+1 ξ m,2 s − ξ 2 r ,2 s , A (2)  rs = max 2 s <n≤2 s+1 ξ 2 r ,n − ξ 2 r ,2 s , A (3)  rs = max uv , say, for some Γ 1 > 0. Using some basic calculation, it follows that (4.12) Similarly, Secondly, since we have (4.17) Kuo-Liang Su 9 Since the first term and the second term in the bracket of (4.17) can be expressed, respectively, as hence, (4.17 Similarly, for some where Hence, for some Combining the results in (4.9), (4.10), (4.19), (4.20), and (4.24), we can conclude that P[lim m,n→∞ ξ mn = 0] = 1.Since P[sup m≥s,n≥t ξ mn > ε] → 0 as s,t → ∞ by applying Lemmas 2.5 and 2.6 on the right-hand side of the previous inequalities, then the proof is completed by a convention that 2 r i=1 Proof of Theorem 3.2.Similar to the previous proof, we start out from assuming that 2 r ≤ m ≤ 2 r+1 and 2 s ≤ n ≤ 2 s+1 with nonnegative integers u and v, and letting First, we can write where (4.27) Comparing (4.9) to (4.24), we can easily obtain that for some and for some Γ 6 ,Γ * 6 > 0, D (1)   rs ≤ Following the arguments in the proof of Theorem 3.1, we can get that for some Γ 7 ,Γ * 7 > 0, The second term in (4.33) is obtained by the following basic calculation.Next, similar to the procedure of getting (4.24), we have where Similarly Then, by the analogous approaches of the proof of Theorem 3.1, for some H > 0, we have Consequently, following the analogous arguments in the previous proof, we can conclude the desired result.
Next, define an array of independent random variables {W ik } with the following properties [8]: Then, it is easy to have that   Then, choosing {X ik } and following the similar steps as in the proof of Theorem 3.3, we can obtain the desired results.

Let {X ik } be an array of orthogonal (in L p (B)) random elements in a Banach space B of type p for some 1 ≤ p ≤ 2 and let {a ik } be any array of real numbers, then there exists positive constants C 3 and C 4 such that, for all m and
is listed below.