Some applications of semimartingale theory to limit theorems

This thesis examines the stochastic convergence of two processes. Firstly a stochastic version ofthe classic Lotka-Volterra model of predatorprey population dynamics is studied. The second process is the empirical process for non-identically distributed random variables. Chapter 2 contains a review of the results on weak convergence. In particular the fundamental results which will be used to prove the theorems in the later chapters are stated. In Chapter 3 the Stochastic Lotka-Volterra model is studied. This chapter extends the paper of Klebaner and Liptser [8], in which they prove the Large Deviations Principle. Here a proof of Functional Central Limit Theorem is given and the associated Moderate Deviations Principle is established. These results results were published in [7]. In the final chapter we study the empirical process for non-identical random variables. The Functional Central Limit theorem for this process was proven by Koul [10]. We present an alternate proof using techniques from semimartingale theory, although we require some extra conditions.