Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have $E[\prod\_{i=1}^n Z\_i^2] = 1$ by independence.


Is there a method for bounding such functions if $X$ was an $n$ dimensional vector where each coordinate is an independent Rademacher $\pm1$ random variable. For instance, for $Z = U^\top X$, the coordinates are not independent anymore. Can one show that $E[\prod\_{i=1}^n Z\_i^2]$ is polynomial in $n$ or $\exp(n)$ even in this setting?


More generally, is there a way of comparing such quantities to the Gaussian setting or universality statements that are applicable in such settings where the random vector may have some weak dependencies?



Comparison of Rademacher and Gaussian moments under linear transformations
By the arithmetic mean--geometric mean inequality and the condition $Z=U^\top X$ for an orthogonal matrix $U$,
$$\prod\_1^n Z\_i^2\le\Big(\frac1n\sum\_1^n Z\_i^2\Big)^n
=\Big(\frac1n\sum\_1^n X\_i^2\Big)^n=1,$$
since $X\_i=\pm1$ for all $i$. So,
$$E\prod\_1^n Z\_i^2\le1,$$
as desired.


