Let $X_1, X_2, \ldots, X_k$ be centered random variables which are not necessarily independent. Under what conditions do we have a bound of the form $$\mathbb{E} \left\lvert X_1 X_2 \cdots X_k \right\rvert \leq C \sqrt{(\mathbb{E} X_1^2)(\mathbb{E} X_2^2) \cdots (\mathbb{E} X_k^2)}$$ for some constant $C > 0$?
- When $k \leq 2$, this follows from Hölder's inequality: $$\mathbb{E} \left\lvert X_1 X_2 \right\rvert \leq \sqrt{(\mathbb{E} X_1^2) (\mathbb{E} X_2^2)}.$$ However, for general $k$, Hölder only gives $$\mathbb{E} \left\lvert X_1 X_2 \cdots X_k \right\rvert \leq \sqrt[k]{(\mathbb{E} X_1^k) (\mathbb{E} X_2^k) \cdots (\mathbb{E} X_k^k)},$$ and in general we do not have $\sqrt[k]{\mathbb{E} X_i^k} \leq \sqrt{\mathbb{E} X_i^2}$.
- This also holds if $X_1, X_2, \ldots, X_k$ are independent. In that case, we have $$\mathbb{E} \left\lvert X_1 X_2 \cdots X_k \right\rvert = (\mathbb{E} \left\lvert X_1 \right\rvert) (\mathbb{E} \left\lvert X_2 \right\rvert) \cdots (\mathbb{E} \left\lvert X_k \right\rvert) \leq \sqrt{\mathbb{E} X_1^2} \sqrt{\mathbb{E} X_2^2} \cdots \sqrt{\mathbb{E} X_k^2}$$
- But it cannot hold in general. Suppose $X_1 = X_2 = \cdots = X_k = X$. Then this inequality says $\mathbb{E} \left\lvert X^k \right\rvert \leq C (\mathbb{E} X^2)^{k/2}$, which would imply that if a random variable $X$ has finite second moments, then it has moments of all orders. This is false: consider a real-valued random variable with probability density function proportional to $\frac{1}{1+x^4}$, for instance.
- The random variable $X$ in the counterexample in the previous bullet point is unbounded. What if we assumed that $X_1, X_2, \ldots, X_k$ were bounded?
Motivation
I'm trying to follow an argument in Bryc, Dembo, and Jiang, "Spectral measure of large random Hankel, Markov and Toeplitz matrices." Near the end of their proof of Proposition 4.12 (page 30 of the arXiv version), they have $$f_n(w) = \sum_{i=1}^q \mathbb{E}\left[ L_w (Q_{a_i} - \tilde{Q}_{a_i}) \prod_{j=1}^{i-1} M_{a_j, a_j} \prod_{j=i+1}^q \tilde{M}_{a_j, a_j} \right],$$ and they write "since the distribution of $(L_w, \{Q_i\}, \{\tilde{Q}_i\})$ is independent of $n > 2k$, while $M_{ii}$ and $\tilde{M}_{ii}$ are normal of mean zero and variance at most $n + 2$, it follows by Hölder's inequality that $\left\vert f_n(w) \right\rvert \leq C_w n^{(q(w) - 1) / 2}$..."
My real question is how this conclusion can be obtained. I can see that based on the information provided, the outer summation and the terms with $L_w$, $Q_i$, or $\tilde{Q}_i$ do not depend on $n$ and can be absorbed into the constant $C_w$. So it remains to study $\mathbb{E}(\prod_{j=1}^{i-1} M_{a_j, a_j} \prod_{j=i+1}^q \tilde{M}_{a_j, a_j})$, which is the expectation of a product of $q-1$ factors. If the inequality at the top of this post holds, then we would have $$\mathbb{E}(\prod_{j=1}^{i-1} M_{a_j, a_j} \prod_{j=i+1}^q \tilde{M}_{a_j, a_j}) \leq \sqrt{\prod_{j=1}^{i-1} \mathbb{E} M_{a_j, a_j}^2 \prod_{j=i+1}^q \mathbb{E}\tilde{M}_{a_j, a_j}^2}.$$ Using $\mathbb{E}M_{a_j, a_j}^2 = O(n)$ and $\mathbb{E}\tilde{M}_{a_j, a_j}^2 = O(n)$, we deduce that the whole expression is $O(n^{(q-1)/2})$, as claimed. However, I'm not seeing the Hölder's inequality argument that would establish the last inequality.