Suppose that $X\_1,\ldots,X\_n$ are independent random variables with $\operatorname E X\_k=0$ and $\operatorname E |X\_k|^p<\infty$ with $1<p<2$ for each $1\le k\le n$. I am interested in the inequalities that establish a lower bound for the $p$-th absolute moment of $S\_n=\sum\_{k=1}^nX\_k$ in terms of the $p$-th absolute moments of $X\_1,\ldots,X\_n$.


I was able to find an upper bound for $E|S\_n|^p$. von Bahr and Esseen (1965) among other results established that
$$
\operatorname E|S\_n|^p\le2\sum\_{k=1}^n\operatorname E|X\_k|^p.
$$
But I can't seem to find an inequality that establishes a lower bound for $\operatorname E|S\_n|^p$. My questions are as follow:



> 
> Are there any inequalities that establish a lower bound for $\operatorname E|S\_n|^p$ in terms of the $p$-th absolute moments of $X\_1,\ldots,X\_n$? Is it true that $\operatorname E|S\_n|^p\ge C\sum\_{k=1}^n\operatorname E|X\_k|^p$ with some positive constant $C$?
> 
> 
> 


Any help is much appreciated!



Lower bound for the $p$-th absolute moment of a sum of random variables
If the $X\_i$ are i.i.d. Gaussian with variance $1$, then you have
$$ c\_p := \mathbb{E} |X\_k|^p = \frac{2^{p/2} \Gamma(\frac{p+1}{2})}{\sqrt{\pi}}.$$
The variable $S\_n$ is also Gaussian with variance $n$, therefore you have
$$\mathbb{E} |S\_n|^p = c\_p n^{p/2}.$$


Hence, $\frac{\sum\_{k=1}^n \mathbb{E} |X\_k|^p}{\mathbb{E} |S\_n|^p} = n^{1-p/2} \rightarrow \infty$ for $1<p<2$. At least, it means that you cannot hope for a constant $C$ as you expected. 


