Are there any general results relating the tail weight of two (or more) probability distributions to the tail weight of their product distribution (in particular, on the assumption that the distributions being multiplied are independent)?
For example: Suppose you have two independent distributions, one Gaussian and the other a Laplace (double-exponential) distribution. My intuition is that their product should have at least exponential tails (i.e., tails at least as heavy as the heavier-tailed of the two distributions being multiplied). Is this true, and if so, how can it be generalized?
(I suspect this might not be a very well-posed question. If so, apologies in advance!)
Yes,this is true. Product of two sub-gaussians is a sub-exponential random variable. The Lemma 2.7.7[1] states that $$ \|XY\|_{\psi_1} \leq \|X\|_{\psi_2} \|Y\|_{\psi_2} $$ It holds even if the random variables are dependent. In particular, a random variable is sub-Gaussian iff its square is sub-exponential (see Lemma 2.7.6[1]).
I think that this result be generalizaed from sub-gaussian distributions to heavier tails by arguing that sufficiently higher-order moments are finite, although I don't have a reference for that right now.
[1]: High-Dimensional Probability, Roman Vershynin. https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.html