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Let $x=(x_1,\ldots, x_n)\in R^n$. Let $r_i, i=1, \ldots, n$ be Rademacher i.i.d. random variables (i.e. $P(r_i=1)=P(r_i=-1)=1/2$). It is a well-known inequality that: $$ E\left(\left|\sum_{i=1}^n x_ir_i\right|^q\right)^{1/q}\leq C\sqrt{q}\|x\|_2 \, . $$

Suppose now that vector $x$ is such that most of coordinates of it are zero and just few $m$ coordinates are equal to $1$. Why in this situation can the inequality above be written as $$ E\left(\left|\sum_{i=1}^n x_ir_i\right|^q\right)^{1/q}\leq C\sqrt m\sqrt{q}\|x\|_2 \, ? $$

Thank you for your help.

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Do you mean exactly $m$ coordinates of $x$ are $1$, and the rest $0$? In this case, $\Vert x\Vert_2=\sqrt m$. The right hand side of the inequality then becomes $C\sqrt m\sqrt q$. –  David Mitra Jan 24 '13 at 22:44

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