I was reading a paper and came across the terms symmetrization and contraction principle of random variables. I tried to extract the statements as follows:
Symmetrization: Let $X_1,\dots,X_n$ be independent zero-mean random variables and $p\geq 2$, then
$$\left\|\sum_{i=1}^n a_i X_i\right\|_p \leq 2 \left\|\sum_{i=1}^n a_i \varepsilon_i X_i\right\|_p$$
where $a_i$ are real numbers and $\varepsilon_i$ denote a sequence of symmetric independent Rademacher random variables (also independent of the $X_i$'s).
Contraction Principle: Let $X_1,\dots,X_n$ be independent non-negative random variables and $p\geq 2$. Further, suppose for each $i$, we have $\mathbb{P}(Y_i\geq t)\geq \mathbb{P}(X_i\geq t)$ for all $t>0$, where $Y_1,\dots, Y_n$ are also non-negative random variables. Then we have
$$\left\|\sum_{i=1}^n a_i \varepsilon_i X_i\right\|_p \leq \left\|\sum_{i=1}^n a_i \varepsilon_i Y_i\right\|_p$$
where $a_i$ are real numbers and $\varepsilon_i$ denote a sequence of Rademacher random variables.
There might be more general statements of these results that exist but the paper does not really cite them, and I am having trouble finding references to exact statements and proofs. If anybody can provide a reference (preferably a textbook) or a hint, that would be greatly appreciated!
For reference, the paper and argument cited is here, on page 12.
Edit: I am looking for a reference to read, not a direct solution or anything like that.
Best Answer
The symmetrization is called Khintchine's inequality, see here: https://en.wikipedia.org/wiki/Khintchine_inequality
For the contraction principle, I don't know any reference, but I can prove it. First note that if $Z\geq 0$ and $p > 0$, then \begin{align} \mathbb{E}[Z^p] = \mathbb{E}\Big[\int_0^{Z} p t^{p-1} dt\Big] = \mathbb{E}\Big[\int_0^{\infty} p t^{p-1} \mathrm{1}_{\{Z\geq t\}} dt\Big] \stackrel{\text{Fubini}}{=} \int_0^{\infty} p t^{p-1} \mathbb{P}(Z\geq t) dt. \end{align} Therefore, if $X,Y\geq 0$ and $\mathbb{P}(X\geq t) \leq \mathbb{P}(Y \geq t)$ for all $t > 0$, then \begin{align*} \mathbb{E}[X^p] = \int_0^{\infty} p t^{p-1} \mathbb{P}(X\geq t) dt \leq \int_0^{\infty} p t^{p-1} \mathbb{P}(Y\geq t) dt = \mathbb{E}[Y^p]. \end{align*} We conclude $(\mathbb{E}[X^p])^{1/p} \leq (\mathbb{E}[Y^p])^{1/p}$ for all $p > 0$.
Thus, with $X = \big|\sum_{i=1}^n a_i \varepsilon_i X_i\big|$ and $Y = \big|\sum_{i=1}^n a_i \varepsilon_i Y_i\big|$, we get the claim.
EDIT: We get the claim assuming $\mathbb{P}(\big|\sum_{i=1}^n a_i \varepsilon_i X_i\big| > t) \leq \mathbb{P}(\big|\sum_{i=1}^n a_i \varepsilon_i Y_i\big| > t)$, which doesn't follow from $\mathbb{P}(X_i > t) \leq \mathbb{P}(Y_i > t), ~1 \leq i \leq n$. Humm, so it's not clear that the third line of the middle equation on page 12 of your reference is actually correct, maybe it was forgotten that the $\varepsilon_i$'s can be negative ? Maybe there is another argument at play, I'm not sure.