Limit theorems for random recurrences and renewal-type processes

Abstract

Random recurrence relations are stochastic difference equations, which define recursively a sequence of random variables in such a way that each element of the sequence is a random function of preceding elements. These equations appear naturally by the analysis of functionals of Markov chains. Understanding the behaviour of solutions to random recurrence relations is important for applications to service systems, networks, insurance processes, processes of coagulation and fragmentation, particle systems and the analysis of randomised algorithms. The thesis is focussed on the asymptotic behaviour of solutions to the recurrence relations associated with certain renewal-type Markov processes including the random walk with barrier, a random occupancy model known as the Bernoulli sieve, and branching random walks. Among the functionals studied are the number of transitions and the absorption time. It is shown that under suitable conditions a wide spectrum of limiting behaviours may appear. The limit distributions may differ from the normal distribution, and the convergence may require nonstandard normalisation.