A Note on the Rate of Strong Convergence for Weighted Sums of Arrays of Rowwise Negatively Orthant Dependent Random Variables

Definition 1. A finite collection of random variables {X_i; 1 ≤ i ≤ n} is said to be NA if for every pair of disjoint subsets A₁ and A₂ of {1,2, …, n},

whenever f₁ and f₂ are nondecreasing functions such that the covariance exists. An infinite collection of random variables {X_i; i ≥ 1} is NA if every finite subcollection is NA.

Definition 2. A finite collection of random variables {X_i; 1 ≤ i ≤ n} is said to be NOD if

for all $$. An infinite collection of random variables {X_i; i ≥ 1} is said to be NOD if every finite subcollection is NOD.

Theorem A. Let {X, X_n; n ≥ 1} be a sequence of identically distributed NA random variables, and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants satisfying

for some 0 < α ≤ 2. Suppose that EX = 0 when 1 < α ≤ 2. If then, for b_n = n^1/α(log⁡⁡n) ^1/γ,

Theorem B. Let {X_ni; i ≥ 1, n ≥ 1} be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants. Assume that there exist some δ with 0 < δ < 1 and some α with 0 < α < 2 such that $$ and assume further that EX_ni = 0 if 1 < α < 2. If for some h > 0 and γ > 0 such that (4), then

where p ≥ 1/α and b_n = n^1/α(log⁡⁡n) ^1/γ.

Theorem C. Let {X_n; n ≥ 1} be a sequence of NOD random variables which is stochastically dominated by a random variable X and let {a_ni; i ≥ 1, n ≥ 1} be a triangular array of real constants such that a_ni = 0 for i > n. Let

where β = max⁡(α, γ) for some 0 < α ≤ 2, γ > 0, and α ≠ γ. Assume that EX_n = 0 for 1 < α ≤ 2 and E|X|^β < ∞. Then, where b_n = n^1/α(log⁡⁡n) ^1/γ.

Definition 3. An array of random variables {X_ni; i ≥ 1, n ≥ 1} is said to be stochastically dominated by a random variable X if there exists a positive constant C such that

for all t ≥ 0, i ≥ 1, and n ≥ 1.

Theorem 4. Let {X_ni; i ≥ 1, n ≥ 1} be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants satisfying $$ for some 0 < α ≤ 2. Assume further that EX_ni = 0 for 1 < α ≤ 2 and E | X|^αlog⁡(1+|X|) < ∞. Then,

where b_n = n^1/α(log⁡⁡n) ^1/α.

Corollary 5. Let {X_n; n ≥ 1} be a sequence of NOD random variables which is stochastically dominated by a random variable X and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants satisfying $$ for some 0 < α ≤ 2. Assume further that EX_n = 0 for 1 < α ≤ 2 and E | X|^αlog⁡(1+|X|) < ∞. Then,

where b_n = n^1/α(log⁡⁡n) ^1/α.

Remark 6. In Theorem 4 and Corollary 5, we consider the case of α = γ for 0 < α ≤ 2 and obtain some strong convergence results for arrays of rowwise NOD random variables and NOD random variable sequences without assumption of identical distribution. The main result settles the open problem posed by Huang and Wang [12]. In addition, it is still an open problem whether

holds true under the same moment condition of Theorem 4.

Lemma 7 (see Bozorgnia et al. [18].)Let {X_i; 1 ≤ i ≤ n} be a sequence of NOD random variables, and let {f_i; 1 ≤ i ≤ n} be a sequence of Borel functions all of which are monotone nondecreasing (or all are monotone nonincreasing). Then, {f_i(X_i); 1 ≤ i ≤ n} is a sequence of NOD random variables.

Lemma 8 (see Asadian et al. [3].)Let M ≥ 2 and let {X_n; n ≥ 1} be a sequence of NOD random variables with EX_n = 0 and E | X_n|^M < ∞ for all n ≥ 1. Then, there exists a positive constant C = C(M) depending only on M such that, for all n ≥ 1,

Lemma 9. Let {X_n; n ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable X. For any u > 0 and t > 0, the following two statements hold:

where C₁ and C₂ are positive constants.

Lemma 10 (see Sung [15].)Let X be a random variable and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants satisfying $$ for some α > 0. Let b_n = n^1/α(log⁡⁡n) ^1/γ for some γ > 0. Then,

Lemma 11 (see Sung [19].)Let X be a random variable and let {a_ni;   1 ≤ i ≤ n, n ≥ 1} be an array of real constants satisfying a_ni = 0 or |a_ni | > 1 and $$ for some α > 0. Let b_n = n^1/α(log⁡⁡n) ^1/α. If q > α, then

Proof of Theorem 4. Without loss of generality, suppose that $$ and a_ni ≥ 0, for all 1 ≤ i ≤ n,   n ≥ 1. For fixed n ≥ 1, define

Denote It is easily seen that, for all ε > 0, which implies that First, we will prove that Actually, for 0 < α ≤ 1, by (14) of Lemma 9, Markov inequality, and E | X|^αlog⁡(1+|X|) < ∞, we have that Next, for 1 < α ≤ 2, by EX_ni = 0, (15) of Lemmas 9 and 10, Markov inequality, and E | X|^αlog⁡(1+|X|) < ∞, we also have that From the above statements, we can get (22) immediately. Hence, for n large enough, To prove (10), it is sufficient to show that It follows from Lemma 10 and E | X|^αlog⁡(1+|X|) < ∞ that For fixed n ≥ 1, it is easily seen that $$ is still a sequence of NOD random variables with mean zero by Lemma 7. Hence, it follows from (14) of Lemmas 9 and 8 and Markov inequality (for M > 2) that

It follows from Lemma 10, (14) of Lemma 9, and Markov inequality that

From Lemma 10 and E | X|^αlog⁡(1+|X|) < ∞, we can obtain that

For fixed n > 1, we divide {a_ni, 1 ≤ i ≤ n} into three subsets {a_ni : |a_ni| ≤ 1/(log⁡⁡n)^m}, {a_ni : 1/(log⁡⁡n)^m < |a_ni| ≤ 1}, and {a_ni : |a_ni| > 1}, where m = (1/(M − α)). Then,

By Lemma 11 and E | X|^αlog⁡(1+|X|) < ∞ again, it follows that Noting that $$, for M > α and fixed n > 1, we have that Noting that $$ and m = 1/(M − α), for M > 2, 0 < α ≤ 2, we have that Finally, we will prove that Hence, by C_r inequality, Markov inequality, Lemmas 9–11, and E | X|^αlog⁡(1+|X|) < ∞, we have that Therefore, the desired result (10) follows from the above statements. This completes the proof of Theorem 4.