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

摘要

Let $1le p2$ and $0lpha , eta infty $ with $1/p=1/lpha +1/eta $. Let ${X_{n}, nge 1}$ be a sequence of random variables satisfying a generalized Rosenthal type inequality and stochastically dominated by a random variable X with $E|X|^{eta } infty $. Let ${a_{nk}, 1le kle n, nge 1}$ be an array of constants satisfying $sum_{k=1}^{n} |a_{nk}|^{lpha }=O(n)$. 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).