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I have Two Independent Gaussian Random Variables $X_1$ and $X_2$ both with zero mean and variances $\sigma_1^2$ and $\sigma_2^2$ respectively. I want to characterize the distribution of the following: $$Z = \int_{0}^{T}X_1(t-\tau)X_2(t)dt$$

I am pretty confident that the mean of $Z$ is zero but I am not sure how to characterize it further. I know that the product of two Gaussians look like the superposition of two Chi Squared random variables. Will the output $Z$ look Gaussian by some type of CLT argument (T can be large if needed)? Any help or advice on this problem would be greatly appreciated!

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    $\begingroup$ Hm, I see a downvote but no comment or answer. Please help me: what should I be thinking now? $\endgroup$ – Gyro Gearloose Dec 9 '15 at 19:36
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    $\begingroup$ I'm not sure the reason for the downvote :( I any pretty stuck on this problem. Characterizing the variance of $Z$ seems non-trivial to me and I am not sure the underlying distribution of $Z$ $\endgroup$ – commEE Dec 9 '15 at 19:55
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    $\begingroup$ By $X_1(t-\tau)X_2(t)dt$ do you mean $f_{X_1}(t-\tau)f_{X_2}(t)dt$? $\endgroup$ – Em. Dec 10 '15 at 7:27
  • $\begingroup$ I would like to try to help, but I'm not sure what you're asking. $\endgroup$ – Em. Dec 10 '15 at 7:27
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This is convolution. See p. 291 here

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