Recursive Methods in Urn Models and First-Passage Percolation

Abstract: This PhD thesis consists of a summary and four papers which deal with stochastic approximation algorithms and first-passage percolation.Paper I deals with the a.s. limiting properties of bounded stochastic approximation algorithms in relation to the equilibrium points of the drift function. Applications are given to some generalized Pólya urn processes.Paper II continues the work of Paper I and investigates under what circumstances one gets asymptotic normality from a properly scaled algorithm. The algorithms are shown to converge in some other circumstances, although the limiting distribution is not identified.Paper III deals with the asymptotic speed of first-passage percolation on a graph called the ladder when the times associated to the edges are independent, exponentially distributed with the same intensity.Paper IV generalizes the work of Paper III in allowing more edges in the graph as well as not having all intensities equal.