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

Let $1\le p<2$ and $0<\alpha , \beta <\infty $ with $1/p=1/\alpha +1/\beta $. Let $\{X_{n}, n\ge 1\}$ be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X with $E|X|^{\beta }< \infty $. Let $\{a_{nk}, 1\le k\le n, n\ge 1\}$ be an array of constants satisfying $\sum_{k=1}^{n} |a_{nk}|^{\alpha }=O(n)$. 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).