Asymptotic Distribution

The present model, and related models, have already received some study [13,19] that suggests that the probability distribution is is Gaussian at long times. The evidence suggest that the distribution is Gaussian for all $\nu<1$ (and there is no reason to believe otherwise). In this chapter, we prove that the distribution function of the normalized position of the walker is asymptotically normal on a restricted domain in parameter space.

For the discussion in this chapter, we will generalize the model slightly by making the bond process discrete in time. If the bond is blocked at time $n$ then, with probability $\alpha$, it will be open at time $n+1$. Likewise, if it is open at time $n$, then with probability $\beta$, it will be blocked at time $n+1$. There is a pair $(\alpha,\beta)$ corresponding to each pair $(p,\nu)$ from the previous chapters. But the converse is not true. There are, for instance, pairs $(\alpha,\beta)$ for which the bond-state correlation oscillates as it decays in time.

To prove the central limit theorem, we analyze the dependence of the future and the past. If they are sufficiently independent then the distribution converges to a normal distribution. We begin by showing that the process is stationary. We can then make use of powerful results from the well-developed theory of stationary processes. In particular we show that our process is uniformly mixing with an exponentially decaying mixing coefficient. (The mixing coefficient is a number that measures the dependence of future and past.) This is a rather strong statement, which satisfies the hypotheses of a number of limit theorems designed for applications.