Expectation of the product of almost independent Gaussians Let $X_ i$ be copies of the standard real Gaussian random variable $X$. Let $b$ be the expectation of $\log |X|$. Assume that the correlations $EX_ iX_ j$ are bounded by $\delta_ {|i-j|}$ in absolute value where $\delta_ k$ is a fast decreasing sequence (in the application I have in mind, $\delta_ k=\exp\{-ck^2\}$ but it should be a huge overkill).
Is it true that there exists a constant $C$ depending on the sequence $\delta_ k$ only such that for all $t_ i>0$, we have $E\prod_ i|X_ i|^{t_ i}\le exp\left( b\sum_ i t_ i+C\sum_ i t_ i^2 \right)$?
 A: As a side note, it seems that we get the opposite inequality for free. If the X_{i} are independent, and we are looking at it for 1 to n, we get
$E[\Pi_{i=1}^{n} \vert X_{i} \vert] = E[\Pi_{i} exp(log(|X_{i}|))] = E[exp(\Sigma_{i} log(|X_{i}|)] \geq exp(E[\Sigma_{i} log(|X_{i}|)]) = exp(nb)$.
Also, you might notice that this doesn't depend at all on the independence of the $X_{i}$... or on the exponent being 1, since we are only taking the expectation of a sum, never a product.
I realize this doesn't answer your original question at all, so I was curious as to where the hypothesis came from. In particular, could you post a proof in that direction when the $X_{i}$ are independnt? Where does C come from?
A: That was extremely difficult to parse.  Until LaTeX support isn't yet enabled, please try to more simply!  (e.g. don't use \left and \right)
Consider the independent case with t constant.  I was hoping C = 0 would work, and the non-zero C only arises because of the dependence.  This isn't the case, however.  By Jensen's inequality:
EΠ|Xi|t = (E|X|t)n = (Eet log|X|)n ≥ etn Elog|X| = etbn.
Thus you need that C > 0 if you want that upper bound.  As you pointed out in reply to unknown's comment, any C will work.  My conjecture is that in the general case with the fast correlation decay, the result should hold for any constant C > 0.
A: Here's my rewording of your question.  Think of Y below as log|X|.
"Let φ(t) = EetY be the moment-generating function of Y.  Suppose that for any C > 0, 
φ(t) ≤ ebt + Ct².  
If Yi are identical copies of Y with fast correlation decay EYiYj ≤ e-a|i-j|, then 
Eet∑Yi ≤ ebnt + Cnt² for all C > 0, 
where the sum is from 1 to n."
