Let $b=(b_1,...,b_n), b_i\in \mathbb R,$ for $i=1,..,n$. Let $\epsilon=(\epsilon_1,..,\epsilon_n)$ be a Rademacher sequence, i.e. $Prob(\epsilon_i=1)=Prob(\epsilon_i=-1)=\frac 12$. It is known that for all $p\geq 2$,

$\left(E|\sum_{i=1}^n\epsilon_ib_i|^p\right)^{1/p}\leq Cp^{1/2} ||b||_2$.

Show similar inequality if in addition:

1) $\sum_{i=1}^n\epsilon_i=0$

2) $\sum_{i=1}^n\epsilon_i=1$

3) $\sum_{i=1}^n\epsilon_i=-1$

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What did you try? – Davide Giraudo Feb 5 '12 at 13:45
I was trying to follow proof in my lecture notes for the inequality without additional assumption. The proof is based just on direct calculation: 1). use multinomial theorem with even p; 2). $E(\epsilon^k)=0$ with even $k$. But in this way I don't see how conditions 1)-3) would change solution... – David Feb 5 '12 at 15:27
I am trying to solve this question. The direct proof in my lecture notes would not work, as now we have dipendent random variable. It turns out that the only problem is to calculate conditional expectation:math.stackexchange.com/q/106533/23993 – David Feb 7 '12 at 3:13