Given
- $X_i$ are independent random variables.
- $|X_i| < 1$
- $E[X_i] = 0$
- $X = \sum_i^n X_i$
- $var(X)=\sigma$
Prove:
$ E(X^p)^{1/p} = O(\sqrt{p}\sigma +p)$ for all even p
Things I've tried:
First note, that all terms with more than p/2 terms are 0 (since $E[X_i] = 0$).
Furthermore, note that $\sum_i X^{4} \leq \sum_i X^2$
So this ends up being some way to count the various terms involving exactly $t$ variables. I don't know how to count this. What should I try?