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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.

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Suppose that all $X_i$ are centered r.v. and $p \geq 2$. Let $S_{a,b} = \sum_{i=a}^b X_i$. We will use Rosenthal's inequality. Recall that the inequality says that

$$A_p \max\left(\sum_{i=a}^b {\mathbb E}|X_i|^p, \left(\sum_{i=a}^b {\mathbb E}X_i^2\right)^{p/2}\right) \leq {\mathbb E}[|S_{a,b}|^p] \leq B_p \max\left(\sum_{i=a}^b {\mathbb E}|X_i|^p, \left(\sum_{i=a}^b {\mathbb E}X_i^2\right)^{p/2}\right),$$ where $A_p$ and $B_p$ are some constants (for $p\geq 2$).

The sum $\sum_{i=1}^\infty X_i$ converges if and only if partial sums form a Cauchy sequence i.e. ${\mathbb E}[|S_{a,b}|^p] \to 0$ as $a \to \infty$. By Rosenthal's inequality, that happens if and only if both series $\sum_{i=1}^\infty {\mathbb E}|X_i|^p$ and $\sum_{i=1}^\infty {\mathbb E}X_i^2$ converge.

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