A Note on the Strong Law of Large Numbers for Arrays of Rowwise ρ̃-Mixing Random Variables

if and only if E|X| < ∞. In the case of independence, Hu and Taylor 1 proved the following strong law of large numbers. Theorem 1.1. Let {Xni : 1 ≤ i ≤ n, n ≥ 1} be a triangular array of rowwise independent random variables. Let {an, n ≥ 1} be a sequence of positive real numbers such that 0 < an ↑ ∞. Let g t be a positive, even function such that g |t| /|t| is an increasing function of |t| and g |t| /|t| 1 is a decreasing function of |t|, respectively, that is,


Introduction
Let {X, X n , n ≥ 1} be a sequence of independent and identically distributed random variables.The Marcinkiewicz-Zygmund strong law of large numbers states that if and only if E|X| α < ∞.In the case of independence, Hu and Taylor 1 proved the following strong law of large numbers.Theorem 1.1.Let {X ni : 1 ≤ i ≤ n, n ≥ 1} be a triangular array of rowwise independent random variables.Let {a n , n ≥ 1} be a sequence of positive real numbers such that 0 < a n ↑ ∞.Let g t be a positive, even function such that g |t| /|t| p is an increasing function of |t| and g |t| /|t| p where k is a positive integer, then Zhu 2 generalized and improved the result of Hu and Taylor 1 for triangular arrays of rowwise independent random variables to the case of arrays of rowwise ρ-mixing random variables as follows.
where v is a positive integer, v ≥ p, then In the following, we will give the definitions of a ρ-mixing sequence and the array of rowwise ρ-mixing random variables.
Let {X n , n ≥ 1} be a sequence of random variables defined on a fixed probability space Ω, F, P .Write F S σ X i , i ∈ S ⊂ N .For any given σ-algebras B, R in F, let ρ B, R sup Define the ρ-mixing coefficients by An array of random variables The ρ-mixing random variables were introduced by Bradley 3 , and many applications have been found.ρ-mixing is similar to ρ-mixing, but both are quite different.Many authors have studied this concept providing interesting results and applications.See, for example, Zhu 2 , An and Yuan 4 , Kuczmaszewska 5 , Bryc and Smole ński 6 , Cai 7 , Gan 8 , Peligrad 9, 10 , Peligrad and Gut 11 , Sung 12 , Utev and Peligrad 13 , Wu and Jiang 14 , and so on.When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.
The main purpose of this paper is to further study the strong law of large numbers for arrays of rowwise ρ-mixing random variables.We will introduce some simple conditions to prove the strong law of large numbers.The techniques used in the paper are inspired by Zhu 2 .

Main Results
Throughout the paper, let I A be the indicator function of the set A. C denotes a positive constant which may be different in various places.
The proofs of the main results of this paper are based upon the following lemma.
As for arrays of rowwise ρ-mixing random variables {X ni : i ≥ 1, n ≥ 1}, we assume that the constant C from Lemma 2.1 is the same for each row throughout the paper.Our main results are as follows.
Theorem 2.2.Let {X ni : i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables and let {a n , n ≥ 1} be a sequence of positive real numbers.Let {g n t , n ≥ 1} be a sequence of positive, even functions such that g n |t| is an increasing function of |t| and g n |t| /|t| is a decreasing function of |t| for every n ≥ 1, respectively, that is, Proof.For fixed n ≥ 1, define , j 1, 2, . . ., n.

2.5
It is easy to check that for any ε > 0, which implies that

2.7
Firstly, we will show that max 1≤j≤n 2.9 which implies 2.8 .It follows from 2.7 and 2.8 that for n large enough, Hence, to prove 2.4 , we only need to show that

2.12
The conditions g n |t| ↑ as |t| ↑ and 2.3 yield that 2.14 which implies 2.12 .This completes the proof of the theorem.

2.15
Theorem 2.4.Let {X ni : i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables and let {a n , n ≥ 1} be a sequence of positive real numbers.Let {g n t , n ≥ 1} be a sequence of nonnegative, even functions such that g n |t| is an increasing function of |t| for every n ≥ 1. Assume that there exists a constant δ > 0 such that g n t ≥ δt for 0 < t ≤ 1.
Proof.We use the same notations as that in Theorem 2.2.The proof is similar to that of Theorem 2.2.
Firstly, we will show that 2.8 holds true.In fact, by the conditions g n t ≥ δt for 0 < t ≤ 1 and 2.16 , we have that 2.20 which implies 2.12 .This completes the proof of the theorem.Proof.In Theorem 2.4, we take

2.23
It is easy to check that {g n t , n ≥ 1} is a sequence of nonnegative, even functions such that g n |t| is an increasing function of |t| for every n ≥ 1.And

2.24
Therefore, by Theorem 2.4, we can easily get 2.4 .
Utev and Peligrad 13, Theorem 2.1 .Let {X n , n ≥ 1} be a ρ-mixing sequence of random variables, EX i 0, E|X i | p < ∞ for some p ≥ 2 and for every i ≥ 1.Then, there exists a positive constant C depending only on p such that .13 which implies 2.11 .By Markov's inequality, Lemma 2.1 for p 2 , C r 's inequality, g n |t| /|t| ↓ as |t| ↑ and 2.3 , we can get that Corollary 2.5.Under the conditions of Theorem 2.4, Let {X ni , i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables and let {a n , n ≥ 1} be a positive real numbers.If there exists a constant β ∈ 0, 1 such that|a n | β |X ni | β < ∞, 2.22then 2.4 holds true.