Limit theorems in reflected Brownian motions and in Markov chains associated with iterated function systems

In this dissertation, we study the limit theorems of two stochastic models. The first is the Markov chain associated to an iterated function system (IFS). An IFS is the name given to a system of maps w = $\{w\sb1, w\sb2, \sp\..., w\sb{N}\}$ on a compact metric space (E, d), such that each $w\sb{i}$ is a contraction, i.e., there exists a constant $r < 1$ such that $d(w\sb{i}(x), w\sb{i}(y)) \le rd(x, y)$ for all $x, y \in E$ and $i = 1, 2, \sp\..., N$. The second is reflected Brownian motion (RBM) a Brownian motion with a lower control barrier at zero (so that the process is prevented from becoming negative); it is used as a tractable model for a flow system in queuing theory and inventory theory. For the Markov chain of an IFS, we prove an ergodic theorem; we first establish a uniform large deviation theorem of the Donsker-Varadhan type, then we prove that there is a unique invariant initial distribution for the Markov chain. Our method is to estimate the value of the relative entropy function of two steady state distributions by comparing it with the rate function of the large deviation theorem. Lipschitz functions play an important role in our estimates. For the RBM, we derive a strong limit theorem for the running maximal of the RBM by using Harrison's computation of the laws of certain functionals of the RBMs and the Vencell-Freidlin estimate of small perturbations of dynamic systems. This result gives the asymptotic growth rate of the "worst case value" of almost every sample path of the RBM's. We also prove some functional central limit theorems associated with the RBM. The methodology is that of the stochastic calculus.