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I would like to calculate or at least to estimate some expectation.

Let $r_k$, $k=1,\ldots, 2n$ be random variables with $P(r_k=1)=P(r_k=-1)=\frac 12$ and such that half of them $r_k=1$ and half $r_k=-1$. Let $b_k$, $k=1,\ldots, 2n$ be real numbers.

I would like to calculate $$ E\left(\prod_{i=1}^n\prod_{j=n+1}^{2n} \exp(r_ib_ir_jb_j)\right). $$

Any ideas would be very helpful.

Thank you.

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    You will need more information about the joint distribution of the $r_k$. For instance, suppose $n=2$ and $b_k = 1$ for $k=1,2,3,4$. One possible model is that you flip a coin; if it is heads you assign the $r_k$ the values (1,1,-1,-1) and if tails (-1,-1,1,1). Then your expectation is $e^{-4}$. Another possible model is that if the coin is heads you take the $r_k$ to be (1,-1,1,-1) and if tails (-1,1,-1,1). Then the expectation is 1.2012-02-22
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    All I know that $r_k$ are Radamacher random variables... Maybe it is possible at least to estimate this expectation?2012-02-22
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    The spelling is Rademacher. Maybe you can find out something about the joint distributions.2012-02-22
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    You'll have to be more specific about what you mean by "estimate". Perhaps some background on why you are interested in this problem would help.2012-02-22
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    I would like to get the upper bound in terms of $b^2$. Well, the background of the problem would not help, as I am working with Fourier analysis...2012-02-22
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    [Here's a cautionary tale](http://math.stackexchange.com/questions/108822/i-need-to-factor-this-function-so-it-is-entirely-dependant-on-x-semicircle-disp/108830#comment254424_108830) about thinking that the background of the problem wouldn't help :-)2012-02-22
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    What is $b$? Your question only has $b_k$.2012-02-22
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    Under $b^2$ I ment the sum over $\sum b_k^2$.2012-02-22
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    Try to use decoupling argument.2012-02-23
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    Sorry, could you please elaborate your answer. I am not familiar with decoupling...2012-02-23

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