Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
2
-
2$\begingroup$ I don’t think this is possible if X and Y are independent. If they are dependent it’s easy of course: just take a Gaussian R and any distribution for Y and then define X=R/Y. $\endgroup$– Carlo BeenakkerCommented Dec 9, 2019 at 22:03
-
2$\begingroup$ How about the Box-Muller transform? The Gaussian-generating interaction can come from how one plays with functions and domains. $\endgroup$– Mickybo YakariCommented Dec 9, 2019 at 22:33
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
7
It was shown in this paper (see formula (2) there) that any normal random variable (r.v.) $Z$ is multiplicatively infinitely divisible; that is, for each natural $k$ there exist iid r.v.'s $W_1,\dots,W_k$ such that $Z$ equals $W_1\cdots W_k$ in distribution; the distribution of $W_1$ is explicitly described.
From that description, it is easy to get an entire continuous one-parameter family $\big((X_t,Y_t)\big)_{t\in(0,1)}$ of pairs $(X_t,Y_t)$ of nontrivial independent r.v.'s such that $X_tY_t$ has the (say) standard normal distribution for each $t\in(0,1)$.
answered Dec 9, 2019 at 22:22
-
$\begingroup$ Thanks, this is very useful. The $W_i$´s seem difficult to construct though. $\endgroup$– rodmsCommented Dec 9, 2019 at 23:07
-
$\begingroup$ @rodms The $W_i$'s are not so difficult to approximate using finite sums, see the Example section of math.stackexchange.com/a/3818660/133843, where the r.v.'s are approximated in Python using the NumPy library. $\endgroup$– WitikoCommented Sep 8, 2020 at 14:06
-
$\begingroup$ @Witiko : Thank you for your comment. $\endgroup$ Commented Sep 8, 2020 at 17:19
-
$\begingroup$ @IosifPinelis Thank you for the result. I will cite your preprint in a work that deals with the initialization of neural network weights, where the product of several iid r.v.'s must have standard normal distribution. $\endgroup$– WitikoCommented Sep 8, 2020 at 17:38
-
$\begingroup$ @Witiko : I am glad this was of help. $\endgroup$ Commented Sep 8, 2020 at 21:06