We consider infinitely divisible distributions with symmetric L\'evy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $\eta(r)=\int_{ |x|\le r} x^2 \nu(dx)$ where $\nu$ is the L\'evy measure, then $\int_0^1 \frac{ r}{ \eta(r)}dr <\infty$ is a sufficient condition for absolute continuity. As far as we know, our result is not implied by existing results about absolute continuity of infinitely divisible distributions.
Categories: math.PR