# Integral of Product of Two Gaussian Variables

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!

• Hm, I see a downvote but no comment or answer. Please help me: what should I be thinking now? – Gyro Gearloose Dec 9 '15 at 19:36
• 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 $Z$ – commEE Dec 9 '15 at 19:55
• By $X_1(t-\tau)X_2(t)dt$ do you mean $f_{X_1}(t-\tau)f_{X_2}(t)dt$? – Em. Dec 10 '15 at 7:27
• I would like to try to help, but I'm not sure what you're asking. – Em. Dec 10 '15 at 7:27