Some problems in probability theory

Thesis

The thesis deals with three problems coming under the heading of probabilistic geometry. They are dealt with in three chapters.

The first chapter is concerned with the angles formed by triples of independent, identically distributed points on a plane. The work is part of an investigation carried out by several workers and motivated by an archaeological problem concerning supposed collinearities formed by ancient sites. In the symmetric Gaussian case the joint distribution is found of two of the angles of the triangle. An integral is given for the density of a named angle in the case of a general Gaussian parent distribution. The asymptotic probability that the triangle has one very obtuse angle is discussed together with the changes that it undergoes when the parent distribution is "stretched". Finally bounds are obtained for this asymptotic probability when the parent distribution is one of a family of circularly symmetric distributions.

The second chapter is about the way in which Brownian motion knots in 3-space. A new concept, that of a "knot-tube", is required because the self-intersecting character of Brownian motion in 3-space means the usual topological notion of a knot can not be applied. It is known that Brownian motion in 5-space and above has very simple topological behaviour, but the 4-space case is unsettled. A partial result is given for 4-space.

The final chapter considers the topological behaviour of contours of two generalisations of Brownian motion to multidimensional time. The two generalisations are the Brownian sheet and the generalisation due to Lévy. Boundedness of the contours of the Lévy process is shown but no analogous result is known for the Brownian sheet. In both cases it is shown that almost surely the union of all non-trivial contours has zero measure. For topological reasons this union cannot be empty.

Authors: Kendall, W

• DOI: 10.5287/ora-w4poqdxmx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford