I have a question about the upper bound of the following moment. Suppose that $(A_1, e_1),\ldots, (A_n, e_n)$ are i.i.d. with $E(e_i)=0$. I am wondering if we have the bound $$E\bigg(\bigg\|\frac{\sum_{i=1}^n A_i}{n} \sum_{j=1}^n e_j\bigg\|^p\bigg)\leq K_p n^{p/2},$$ where $K_p$ is a constant free of $n$. We assume $E(\|A_i\|^p \|e_i\|^p)<\infty$.
I know if $A_i$ is a constant, the desired result holds by Marcinkiewicz–Zygmund inequality.