| Description |
This dissertation contains the solutions to two problems. The first problem concerns probability on trees. The second is a covering problem. Several different probability problems can be studied on trees, such as random walks, percolation, random-walk-in-random-environment and tree-indexed processes. Optimal-reward problems are one reason to study tree-indexed processes. We assign i.i.d. Gaussian random variables with mean zero and variance one to the edges of a tree. We show that if the growth of the tree has been restrained, then the asymptotic behavior of optimal rewards can be characterized. Since the weights are Gaussian random variables, the optimal reward at depth n can be treated as the supremum of a Gaussian process. To find the asymptotic behavior of optimal rewards, we appeal to Borell's inequality and metric entropy. If i.i.d. random variables (not necessarily Gaussian) are attached to the edges of a spherically-symmetric tree, then we prove that the sooner the tree branches more, the greater the value of the optimal reward. Covering problems were first considered on the one-dimensional torus and a complete characterization of coverage was given by Shepp in 1972. The one-dimensional covering problem has wide variant problems studied through the past decades. The covering problem that we consider is about covering a simple connected curve on the two-dimensional torus with squares. A method of projection and the martingale convergence theorem will be used in our analysis. We found a necessary and sufficient condition for coverage that is similar to Shepp's condition. |
