Let $X$ and $Y$ be two real random variables with smooth densities $m_X$ and $m_Y$.
- Is there a natural condition on $m_X$, $m_Y$ and the dependencies of $X$ and $Y$ such that $(X, Y)$ has a joint density $p(x, y)$ in $L^2(\mathbb{R}^2)$?
- What upperbound can we expect for the norm of $p$?
Here are some specific cases:
- if $X = Y$, then the joint distribution doesn't have a density.
- if $X$ is Gaussian and $Y = (2B- 1)H$ with $B$ an independent Bernouilli, then $X$ and $Y$ are uncorrelated but the joint distribution doesn't have a density.
- if $X$ and $Y$ are independent then $|p|_{L^2} = |m_X|_{L^2}|m_X|_{L^2}$
Thanks.