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](https://arxiv.org/pdf/0904.4723.pdf), on page 12.

Edit: I am looking for a reference to read, not a direct solution or anything like that.