# Comparing $L_p$ norms of sums of Gaussians and Bernoulli random variables

Problem:

Let $$(\epsilon_i), (g_i)$$ be sequences of independent Bernoulli$$(\{1,-1\},0.5)$$ and Gaussian$$(0,1)$$ random variables respectively (the sequances are also independent from each other).

Show that $$\| \sum_{i=1}^n g_ia_i\|_{L_p} \ge \frac{\sqrt{2}}{\sqrt{\pi}} \| \sum_{i=1}^n \epsilon_i a_i\|_{L_p}$$ for any $$p \ge 1$$ and any fixed real sequence $$(a)$$ and fixed $$n \in \mathbb N.$$

Attempt: The hint is to start by showing that $$\epsilon_i |g_i|$$ is also a standard Gaussian random variable which is reasonably doable. Furthermore noticing that $$\sqrt{2/\pi}= \|g_i\|_{L_1}$$ we have $$\| g_N \|_1 \| \sum a_i\epsilon_i\|_{L_p} =\left[(\int |g_N|)^p(\int| \sum a_i \epsilon_i|^p) \right]^{1/p}$$ and $$\left[(\int |g_N|)^p(\int| \sum a_i \epsilon_i|^p) \right]^{1/p} \overbrace{\le}^{Jensen} \left[(\int |g_N|^p)(\int| \sum a_i \epsilon_i|^p) \right]^{1/p} \overbrace{=}^{independence} \left[\int (|g_N|\times| \sum a_i \epsilon_i|)^p \right]^{1/p}$$ but carrying on from here seems difficult because we're losing the independence at the last step.

Context:

The goal is to show that the Khinchine inequalities (top of page 14 in the source) for Bernoulli random variables (ie the $$\psi_2$$ character of the finite sum $$\sum_1^n \epsilon_i a_i$$ for an arbitrary real sequence $$(a)$$ and IID random variables $$\epsilon_i$$ following the Bernoulli law on $$\{1,-1\}$$ with parameter $$p=1/2$$). One known way to do this is via the Hoeffding inequality.

However here we want to do it via the $$\psi_2$$ character of $$\sum_1^n a_i g_i$$ for $$g_i$$ iid standard gaussian variables.

Let $$(\epsilon_i), (g_i)$$ be sequences of independent Bernoulli$$(\{1,-1\},0.5)$$ and Gaussian$$(0,1)$$ random variables (the sequances are also independent from each other).

Q1. Show that $$\epsilon_i |g_i|$$ is also a standard Gaussian random variable.

This question is relatively simple and follows from the independence and the fact that it is enough to check that $$P(\epsilon_i |g_i|>t)= P(g_i>t)$$ for all $$t$$.

Q2. Show that $$\| \sum_{i=1}^n g_ia_i\|_{L_p} \ge \frac{\sqrt{2}}{\sqrt{\pi}} \| \sum_{i=1}^n \epsilon_i a_i\|_{L_p}$$ for any $$p \ge 1$$ and any real sequence $$(a)$$. Deduce the Khinchine inequalities.

For the first part, I tried to remark that by independence $$\| \sum_{i=1}^n \epsilon_ia_i\|_{L_p} \| \sum_{i=1}^n f(g_i)\|_{L_p} =\| (\sum_{i=1}^n \epsilon_ia_i)( \sum_{i=1}^n f(g_i))\|_{L_p}$$ for $$f(.) = |.|$$ but I don't see how to proceed from here.

Solution for the deduction: It is enough to establish that for all $$p\ge 1$$ we have control over the $$p$$ norms $$\| \sum_{i=1}^n \epsilon_i a_i\|_{L_p} \le c p^{1/2}$$ for some $$c>0$$ which directly follows from the $$\psi_2$$ character of $$\sum_{i=1}^n g_i a_i$$ and the initial inequality.

A hint on how to carry on would be more than welcome.

As you said, suppose $$\lVert\sum g_i a_i\rVert_{L_p} = \lVert\sum a_i \varepsilon_i |g_i| \rVert_{L_p}.$$
\begin{align} \|\sum g_i a_i\rVert_{L_p} &= \|\sum a_i \varepsilon_i |g_i| \|_{L_p} \\ \text{(by independence)}&=\mathbb E_{\varepsilon} \left[\mathbb E_g \left(\left|\sum a_i \varepsilon_i |g_i| \right|^p\right)\right]^{1/p}\\ (\text{by Jensen on }x \mapsto |x|^{p})&\ge \mathbb E_{\varepsilon} \left[ \left|\mathbb E_g \left(\sum a_i \varepsilon_i |g_i| \right)\right|^p\right]^{1/p} \\ &=\sqrt{2/\pi} \cdot \mathbb E_\varepsilon\left(\left|\sum a_i \varepsilon_i\right|^p \right)^{1/p} \end{align}