Let $X_1, X_2, \ldots$ be independent $L_p$ random variables. I'm looking for useful conditions which imply $S_n = \sum_{i = 1} ^ n X_i$ converges in $L_p$ to some random variable $S$. If it is helpful, $X_i$ can be assumed symmetric without loss of generality, and interest is primarily in $p > 2$. For $1 \le p \le 2$ we can get upper bounds on $E|S_n|^p$ of the form $C_p\sum_{i = 1} ^ n E|X_i| ^ p$ which is useful, but nothing like that works for $p > 2$.
This may be a bit vague, mainly because I'm not entirely sure what I'm looking for. I suppose the essence is this: If one wanted to show that $S_n \to S$ in $L_p$ ($p > 2$ emphasized), where $S_n$ is a sum of independent random variables what would one try? Probably anything that gives a general method for doing this that is more substantive than "check $S_n$ is Cauchy wrt $\|\cdot\|_p$" would be useful.