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

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 n k , 1 ≤ k ≤ n , n ≥ 1 } be an array of constants satisfying ∑ k = 1 n | a n k | 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 (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 ).