Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:


If $\{S\_i, \mathcal{F}\_i, 1\leq i \leq n\}$ is a martingale and $2\leq p < \infty$, then there exist constants $C\_1$ and $C\_2$ depending only on $p$ such that 


$$
\begin{align}
C\_1\Bigg\{E\left[\left(\sum\_{i=1}^{n} E(X\_i^2 | \mathcal{F}\_{i-1})\right)^{p/2}\right]& + \sum\_{i=1}^{n}E |X\_i|^p\Bigg\}\\& \leq E|S\_n|^p \leq\\& C\_2 \Bigg\{ E\left[\left(\sum\_{i=1}^{n}E(X\_i^2|\mathcal{F}\_{i-1})\right)^{p/2}\right] + \sum\_{i=1}^{n}E |X\_i|^p\Bigg\}
\end{align}
$$


where $X\_i = S\_i - S\_{i-1}$. I am more interested in the sum of iid case, whereby $S\_k = \sum\_{i=1}^{k} X\_i$ and $\{X\_1,...,X\_n\}$ are iid random variables. In this case, $E(X\_i^2 | \mathcal{F}\_{i-1}) = E(X\_i^2)$. 


$\textbf{QUESTION:}$ I want to know if there is a conditional version of this inequality floating around in literature somewhere, as I cannot seem to find anything. In particular, in the case of iid random variables, and looking at sums of these rv's, if $\mathcal{G}$ is a $\sigma$ - sub algebra of $\mathcal{F}\_n$, can I say something like this:


$$
\begin{align}
C\_1\Bigg\{\left(\sum\_{i=1}^{n} E(X\_i^2 | \mathcal{G})\right)^{p/2}& + \sum\_{i=1}^{n}E (|X\_i|^p|\mathcal{G})\Bigg\}\\& \leq E(|S\_n|^p | \mathcal{G}) \leq\\& C\_2 \Bigg\{ \left(\sum\_{i=1}^{n}E(X\_i^2|\mathcal{G})\right)^{p/2} + \sum\_{i=1}^{n}E (|X\_i|^p|\mathcal{G})\Bigg\},
\end{align}
$$


almost surely?


EDIT: I should note that $\mathcal{F\_k} = \sigma(X\_1,...,X\_k)$, the sigma algebra generated by $X\_1,...,X\_k$ in the iid setting.



Conditional Form of Rosenthal's Inequality
The answer depends on $\mathcal G$, even for independent sequences. 


* If $\mathcal G$ is independent of $\mathcal F\_n$, then this reduces to the classical Rosenthal's inequality.
* However, if each $X\_i$ is $\mathcal G$-measurable, then the problem reduces to determine whether we can find a constant $C$ depending only on $p$ such that for each $n$,
$$C^{-1}\left(\left(\sum\_{i=1}^n X\_i^2\right)^{p/2}+\sum\_{i=1}^n|X\_i|^p\right)\leqslant 
|S\_n|^p\leqslant C\left(\left(\sum\_{i=1}^nX\_i^2\right)^{p/2}+\sum\_{i=1}^n|X\_i|^p\right)\mbox{ a.s.}$$
If it was true, then we would have in particular that 
$$\tag{\*}|S\_n|^p\leqslant C\left(\left(\sum\_{i=1}^nX\_i^2\right)^{p/2}+\sum\_{i=1}^n|X\_i|^p\right)\mbox{ a.s.}$$
If we take $X\_i=\pm 1$ with probability $1/2$, then $|S\_n|^p=n^p$ on a set of probability $2^{-n+1}$ (hence of positive probability) while the RHS of (\*) is $C(n^{p/2}+n$).


There exists a conditional Rosenthal-type inequality, in [Conditional limit theorems for conditionally negatively associated random variables](http://link.springer.com/article/10.1007%2Fs00605-010-0196-x) by De-Mei Yuan, Jun An and Xiu-Shan Wu. It works for random variables which are conditionally negatively associated with respect to $\mathcal G$.


