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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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    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:http://math.stackexchange.com/q/106533/239932012-02-07

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