From sojourn times and boundary crossings to iterated random functions

Abstract

This thesis was motivated by related sojourn time and boundary crossing problems for the Brownian motion process. In Chapter 1, we discuss these motivating problems, and outline how they led naturally to the research in subsequent chapters. The available literature relevant to this thesis is reviewed in Chapter 2.

In Chapter 3, we extend the uniform law for sojourn times of processes with cyclically exchangeable increments to the case of random fields, with general parameter sets, that possess a suitable invariance property.

In Chapter 4, we consider several special cases of iterations of random i.i.d. linear functions with beta distributed fixed points that describe a nested interval scheme when iterated in a backward direction, and an ergodic Markov chain in the forward direction. We find these points by extending a result due to E. J. G. Pitman (1937), which allows us to solve the random equation by solving an equivalent random equation with a gamma distributed solution. In each case, these fixed points are the almost sure limit of the corresponding nested interval scheme and the stationary distribution of the corresponding Markov chain. Using our new technique, we also derive some new special cases of randomly weighted averages, and show how our results define some new stick-breaking constructions of beta distributed random variables.

In Chapter 5, we consider a class of discrete time Markov chains with state space [0,1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

In Chapter 6, using our extension of the result due to Pitman (1937) from Chapter 4, we characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n-1)···X(1) of i.i.d. copies X(k) of X converge a.s. as n approaches infinity and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

In Chapter 7, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for “close boundaries” in the univariate case. We also obtain upper bounds for the first boundary crossing time densities.