Strong laws for weighted sums of random variables satisfying generalized Rosenthal type inequalities

Abstract

Abstract Let 1 ≤ p < 2 $1\le p<2$ and 0 < α , β < ∞ $0<\alpha , \beta <\infty $ with 1 / p = 1 / α + 1 / β $1/p=1/\alpha +1/\beta $ . Let { X n , n ≥ 1 } $\{X_{n}, n\ge 1\}$ be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X with E | X | β < ∞ $E|X|^{\beta }< \infty $ . Let { a n k , 1 ≤ k ≤ n , n ≥ 1 } $\{a_{nk}, 1\le k\le n, n\ge 1\}$ be an array of constants satisfying ∑ k = 1 n | a n k | α = O ( n ) $\sum_{k=1}^{n} |a_{nk}|^{\alpha }=O(n)$ . Marcinkiewicz–Zygmund type strong laws for weighted sums of the random variables are established. Our results generalize or improve the corresponding ones of Wu (J. Inequal. Appl. 2010:383805, 2010), Huang et al. (J. Math. Inequal. 8:465–473, 2014), and Wu et al. (Test 27:379–406, 2018).

Keywords

• Strong law of large numbers
• Weighted sum
• Widely orthant dependent random variable