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if we consider an independent sequence of random variables $X_1,X_2,...$ and denote with $S_n = \sum_{j=1}^n X_j$ the sequence of partial sums, one can easily verify the Marcinkiewicz-Zygmund inequality:

$\Vert S_n \Vert_2^2 \leq \sum_{j=1}^n \Vert X_j \Vert_2^2$.

I need such a result where $(X_n)_{n \in \mathbb{N}}$ is a sequence of martingales differences or in other words, where $S_n$ is a sequence of martingales. But somehow I manage to find only results of the type

$\Vert S_n \Vert_p^2 \leq c_p \sum_{j=1}^n \Vert X_j \Vert_p^2$

for either $p>2$ or $1<p<2$. Is the case for $p=2$ that trivial?

student7481
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  • For $p=2$, $$ \mathsf{E}S_n^2=\sum_{j=1}^n \mathsf{E}X_j^2. $$ –  Mar 29 '22 at 20:52
  • But does such a result also hold true if $X_n$ is not an independent sequence of random variables? – student7481 Mar 29 '22 at 21:07
  • It suffices to assume that $X_j$'s are uncorrelated (and $\mathsf{E}X_j=0$, $j=1,\ldots,n$). –  Mar 29 '22 at 21:08
  • And if I can only assume, that $S_n$ is a sequence of martingales? Without any further correlation assumptions – student7481 Mar 29 '22 at 21:21
  • Well, an MDS satisfies that condition trivially (i.e., the increments of a martingale are uncorrelated). –  Mar 29 '22 at 21:22
  • Thanks a lot, I wasn't aware of this fact. (https://math.stackexchange.com/questions/1257086/uncorrelated-successive-differences-of-martingale) – student7481 Mar 29 '22 at 21:30
  • In general, for an $L^p$-bounded MDS with $1\le p\le 2$, $$ \mathsf{E}|S_n|^p\le 2\sum_{j=1}^n \mathsf{E}|X_j|^p. $$ –  Mar 29 '22 at 21:32

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