Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on $\mathbb{ R}_{ +}=[0,\infty)$, we prove that their free multiplicative convolution $\mu\boxtimes\nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{ -1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty)$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu\boxtimes\nu$ is another Jacobi measure whose density has square root decay at the edges of its support.
Categories: math.PR