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Say we have a sequence $\{X_i\}$ independent which is mean $0$ and $E|X_i|^p < \infty$ for $p \geq 1$. Is there a bound in the form $E\left|\sum_{i=1}^n X_i\right|^p \leq C\cdot \sum_{i=1}^n E| X_i|^p$ where $C$ is not function of $n$?

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No. For example, if $X_i$ are standard normal, $\sum_{i=1}^n X_i$ is normal with mean 0 and variance $n$, so $E\left|\sum_{i=1}^n X_i \right|^p = K(p) n^{p/2}$ where $K(p) = E |Z|^p$ for a standard normal random variable. But the right side is $n C K(p)$. So no such bound can exist if $p > 2$.

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Thanks for the quick reply. Could you say something if there is more structure in the sequence? If, for instance, $\{X_i\}$ are centered exponentials with $X_i \sim Exp(i)$ (with mean removed) or if $\{X_i\}$ more generally are either upper or lower bounded, could anything be said about a sum of expectation bound? – monte Dec 8 '11 at 19:21
Consider e.g. $p=4$. Then $E[ |\sum_{i=1}^n X_i|^4 ] = n E[X_i^4] + 3 n(n-1) E[X_i^2]^2 \sim K n^2$ as $n \to \infty$. Similarly, for any even integer $p$ the left side of your inequality should be on the order of $n^{p/2}$ as $n \to \infty$. – Robert Israel Dec 8 '11 at 20:17

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