Remarkable asymmetric random walks

There exists an asymmetric probability measure $P$ on the real line with $P^{\ast n} (]0,\infty[) + (1/2) P^{\ast n} ({\left\{ 0 \right\}}) = 1/2$ for every $n \in \mathbf{N}$. $P$ can be chosen absolutely continuous and $P$ can be chosen to be concentrated on the integers. In both cases, $P$ can be chosen to have moments of every order, but $P$ cannot be determined by its moments.