Asymptotics for Voronoi tessellations on random samples

Let V(X1,…,Xn) denote the total edge length of the Voronoi tessellation on random variables X1,…,Xn. If X1,X2,… are independent and have a common continuous density f(x) on the unit square which is bounded away from 0 and ∞ then it is shown thatlimn→∞V(X1,…,Xn)n1/2=2∫[0,1]2(f(x))1/2 dx c.c.,where c.c. denotes complete convergence.