Conditional Form of Rosenthal's Inequality

Question

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.

Answer 1

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 by De-Mei Yuan, Jun An and Xiu-Shan Wu. It works for random variables which are conditionally negatively associated with respect to $\mathcal G$.