Given a finitely supported probability measure on a Kleinian group, Kaimanovich showed that the Poisson boundary of the associated random walk is the boundary of the hyperbolic space equipped with the hitting measure. It has been conjectured that the hitting measure is singular to conformal measures of the Kleinian group. In this talk, we mainly focus on how Mostow’s rigidity can be generalized to show the expected singularity when the Kleinian group is not convex cocompact, which is my joint work with Andrew Zimmer. We also discuss other applications of this general machinery besides the singularity conjecture.