Month: December 2010

If $X$ and $Y$ are independent real random variables with densities $f$ and $g$ then $X+Y$ has density  $f*g$. This density is bounded as soon as $f$ or $g$ is bounded since $$\left\Vert f*g\right\Vert_\infty\leq \min(\left\Vert f\right\Vert_\infty,\left\Vert g\right\Vert_\infty).$$ One may ask if $XY$ has similarly a bounded density. The answer is  unfortunately negative in general. To see it, we note first that when $X$ and $Y$ are non negative then $XY$ has density $$t\in\mathbb{R}_+\mapsto \int_0^\infty\!\frac{f(x)}{x}g\left(\frac{t}{x}\right) dx.$$ Now if  for instance $X$ and $Y$ are uniform on $[0,1]$ then $XY$ has density $$t\mapsto -\log(t)\mathbf{1}_{[0,1]}(t)$$ which is unbounded... If $X$ is  non negative with density $f$ then $X^2$ has density $$t\in\mathbb{R}_+\mapsto \frac{f(\sqrt{t})}{2\sqrt{t}}.$$ For instance when $X$ is uniform then $X^2$ has (unbounded) density $$t\mapsto \frac{1}{2\sqrt{t}}\mathbf{1}_{[0,1]}(t).$$ The density of $X^2$ is bounded if $f$ is bounded and $f(t)=O(t)$ as $t\to0$ (imposes $f(0)=0$).