The central limit theorem in a random environment

Abstract

In this paper we started by explaining what a Markov chain is. After this we defined some key concepts such as stationarity, reversibility and ergodicity which were used throughout the rest of the paper. Next, the classical central limit theorem was stated in order to refresh the reader of this theorem and to show that, because of the dependence, this theorem is not applicable for the Markov chains we discussed earlier. In the next section we started working on the Kipnis-Varadhan central limit theorem. We defined the Martingale central limit theorem and used it to show that the additive functions can be approximated by a martingale. This had the result that the Martingale central limit theorem implies the central limit theorem for the additive functions. Furthermore, we also derived an equation for calculating the limiting variance. For this we used the key concepts discussed in section 2 and another concept called spectral measures. The section is finished by a simple example in order to illustrate what we have discussed previously. In the next section we used simulations in one-dimension and two-dimensions to show that, in a random environment, the distribution of the additive functions indeed converges to that of the normal distribution. Because the probability distributions defined on the random environment was a symmetric distribution the Kipnis-Varadhan central limit theorem was not yet needed since the Martingale central limit theorem already applied. After this we defined some key concepts for a Markov chain working in continuous time. The Kipnis-Varadhan central limit theorem was proven for these Markov chains as well. Furthermore, a different equation was derived to find the limiting variance for continuous time Markov chains. In section 7 the random conductance model was explained. In this chapter the random conductance model was used in order to make simulations to show that the Kipnis-Varadhan central limit theorem works. After simulating the Markov chain in continuous time we indeed saw that the histograms show convergence towards the normal distribution. This shows that for the simulations the Kipnis-Varadhan central limit theorem works.