For the perimeter length Ln and the area An of the convex hull of the first n steps of a planar random walk, this thesis study n â mean and variance asymptotics and establish distributional limits. The results apply to random walks both with drift (the mean of random walk increments) and with no drift under mild moments assumptions on the increments. Assuming increments of the random walk have finite second moment and non zero mean, Snyder and Steele showed that nâ1Ln converges almost surely to a deterministic limit, and proved an upper bound on the variance Var[Ln] = O(n).We show that nâ1Var[Ln] converges and give a simple expression for the limit,which is nonzero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the nondegenerate case.Then we focus on the perimeter length with no drift and area with both drift and zerodrift cases. These results complement and contrast with previous work and establish nonGaussian distributional limits. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are nonzero.
Date of Award  1 Apr 2015 

Language  English 

Awarding Institution   University Of Strathclyde


Sponsors  EPSRC (Engineering and Physical Sciences Research Council) & University of Strathclyde 

Supervisor  Xuerong Mao (Supervisor) & Andrew Wade (Supervisor) 
