We already have results from Tropp's book (https://arxiv.org/pdf/1501.01571.pdf), theorem 1.6.2:
Let $S_1,...,S_n$ be independent, centered random matrices with common dimension $d_1 \times d_2$, and assume that each one is uniformly bounded
$$\mathbb{E}S_k = 0\ \text{ and }\ \|S_k\| ≤ L\ \text{for each}\ k = 1,...,n$$
Let
$$Z = \sum_{i=1}^n S_k $$
and let $v(Z)$ denote the matrix variance statistic of the sum:
$$v(Z) = \max\{\mathbb{E}(ZZ^*),\mathbb{E}(ZZ^*)\}$$
Then $$\mathbb{E}\|Z\| \leq (2v(Z)\ln(d_1+d_2))^{0.5}+\frac{1}{3}L\ln(d_1+d_2) .$$
Can we establish a similar bound for $\mathbb{E}\|Z\|^3$, using $v(Z), d_1, d_2, L$?
PS: all norms appear above are 2-norms.
The answer is in https://arxiv.org/pdf/1109.1637.pdf by Tropp:
$$\Big[\mathbb{E}\|\sum_{i=1}^n Y_i\|_2^q\Big]^{\frac{1}{q}}\leq \sqrt{er}\Big\|\Big(\sum_{i=1}^n\mathbb{E} Y_i^2 \Big)^{\frac{1}{2}}\Big\|_2+2er\big(\mathbb{E} \max_i \|Y_i\|_2^q\big)^{\frac{1}{q}}$$