Assuming $X\sim N(\mu_1, \sigma_1^2)$, $Y\sim N(\mu_2,\sigma_2^2)$, and $\operatorname{cov}(X, Y) = c$, is this possible to get a tail bound for $XY$, which I mean $P(XY \geq t)$? As first, I am thinking about if $XY$ is sub-Gaussian or sub-exponential but it seems to be not true. Thanks.
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1The distribution of $XY$ is known and can be found here: http://math.stackexchange.com/a/397716/248958 And so calculating the probability is a doable task. – user6291 Oct 29 '15 at 00:40