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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?

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Where does this question come from? –  Igor Rivin Aug 22 '12 at 15:41
1  
Look at Lemma 4, page 172, in Billingsley's book Convergence of probability measures. Lemma is for p=4, but it works for all even p. If I am not mistaken, this lemma gives the bound you are looking for. –  user21541 Aug 22 '12 at 16:08
    
@waler: Can you identify which section that's from? In my copy of the second edition of that book, there is no Lemma 4 on or near page 172. –  Mark Meckes Aug 22 '12 at 18:04
    
@Mark Meckes: 1968 edition, chapter 4. dependent variables, section 20: mixing processes, paragraph on moment inequalities. –  user21541 Aug 22 '12 at 18:40
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1 Answer

Take a loot at R. Ibragimov and Sh. Sharakhmetov, The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero, Theory Probab. Appl., 46(1), 127–132. (6 pages) Read More: http://epubs.siam.org/doi/abs/10.1137/S0040585X97978762

Abstract Let $\xi_1, \ldots, \xi_n$ be independent random variables with ${\bf E}\xi_i=0,$ ${\bf E}|\xi_i|^t<\infty$, $t>2$, $i=1,\ldots, n,$ and let $S_n=\sum_{i=1}^n \xi_i.$ In the present paper we prove that the exact constant ${\overline C}(2m)$ in the Rosenthal inequality $$ {\bf E}|S_n|^t\le C(t) \max \Bigg(\sum_{i=1}^n{\bf E}|\xi_i|^t,\ \Bigg(\sum_{i=1}^n {\bf E}\xi_i^2\Bigg)^{t/2}\Bigg) $$ for $t=2m,$ $m\in {\bf N},$ is given by $$ \overline C(2m)=(2m)! \sum_{j=1}^{2m} \sum_{r=1}^j \sum \prod_{k=1}^r \frac {(m_k!)^{-j_k}} {j_k!}, $$ where the inner sum is taken over all natural $m_1 > m_2 > \cdots > m_r > 1$ and $j_1, \ldots, j_r$ satisfying the conditions $m_1j_1+\cdots+m_rj_r=2m$ and $j_1+\cdots+j_r=j$. Moreover $$ \overline C(2m)={\bf E}(\theta-1)^{2m}, $$ where $\theta $ is a Poisson random variable with parameter 1.

In your case, one gets $$ {\bf E}|S_n|^{2m} \le \overline C(2m) \max( n, (var S_n)^{m})= \overline C(2m) \max( n, n^{m}c^m). $$ where $c=var(\xi_i)$. Thus for large $m$, $$ \Bigl( {\bf E}|S_n|^{2m}\Bigr)^{\frac 1{2m}} \le (\overline C(2m))^{1/(2m)} \sqrt{n} c. $$ Finally, they cite earlier papers where it was shown that $$ \overline C(t)= O( t/\log t). $$ Ans since is $(t/\log t)^{1/t}<\sqrt{t}$ for $t>2$ we are done.

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That is a nice paper but is overkill for what the OP asks. For the $t/ \log t$ result, see the first paper by Schechtman, Zinn, and myself. It also contains an easy proof of the bound $t/ (\log t)^{1/2}$. –  Bill Johnson Aug 22 '12 at 17:11
    
Johnson, W. B.; Schechtman, G.; Zinn, J. Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 (1985), no. 1, 234–253. –  Bill Johnson Aug 22 '12 at 17:14
    
@Bill Johnson: absolutely, severe overkill. result must follow also from some easier inequalities as well. –  user21541 Aug 22 '12 at 18:36
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