In this paper, the complete convergence for the weighted sums of independent and identically distributed random variables in Stout [9] is improved and extended under NOD setup.The more optimal moment condition is given. The main results also hold for END sequence. §1 Introduction Theorem 4.1.4 (i) of Stout [9] showed the following complete convergence for Stout's type weighted sums of independent and identically distributed random variables.be a sequence of independent and identically distributed random variables, {a nk , n ≥ 1, k ≥ 1} an array of constants with sup k≥1 |a nk | ≤ Kn −α (1) for some K > 0 and sup n≥1 c n ≤ Kn β−α (2) for some β > −(1 + α), where c n = ∞ k=1 a 2 nk . If (1 + α + β)/α > 2 and ∞ n=1 exp{−u/c n } < ∞ (3)for all u > 0, then EX = 0 and E|X| (1+α+β)/α < ∞ imply that for any ε > 0The concept of complete convergence was introduced by Hsu and Robbins [6] and they proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite.

Let 1 ≤ p < 2 and 0 < α, β < ∞ with 1/p = 1/α + 1/β. Let {X n , n ≥ 1} be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X with E|X| β < ∞. Let {a nk , 1 ≤ k ≤ n, n ≥ 1} be an array of constants satisfying n k=1 |a nk | α = O(n). Marcinkiewicz-Zygmund type strong laws for weighted sums of the random variables are established. Our results generalize or improve the corresponding ones of Wu (

For a simple linear errors-in-variables regression model with widely orthant dependent errors, we provide sufficient conditions for the convergence rate in the strong consistency of the least squares estimators. We also provide necessary conditions. Our result improves and extends some results of Liu et al. (J. Math. Ineq., 14 (2020), 771-779).

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