The convex hull of samples from self-similar distributions. Advances in Applied Probability, 31, 34-47 (1999)

Let X₁, X₂, X₃,... be i.i.d. random points in R² with distribution µ, and let N_n denote the number of points spanning the convex hull of X₁, X₂,..., X_n. We obtain $\liminf_{{n -> \infty}}E N_n n^-1/3 \leq \gamma₁$ and $E N_n \leq \gamma₂ n^1/3 (\log n)^2/3$ for some positive constants $\gamma₁, \gamma₁$ and sufficiently large n under the assumption that µ is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in R^d.