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Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class of random variables is that it leads to the functional CLT without the need to estimate rates of convergence of mixing coefficients. More precisely, because of the maximal moment inequalities established in the previous chapter, we are able to prove tightness for a stochastic process constructed from a negatively dependent sequence. Furthermore, we establish the convergence of the partial sums process, either to a Gaussian process with independent increments or to a diffusion process with deterministic time-varying volatility. We also provide the multivariate form of these functional limit theorems. The results are presented in the non-stationary setting, by imposing Lindeberg’s condition. Finally, we give the stationary form of our results for both asymptotic positively and negatively associated sequences of random variables.

Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter introduces the necessary mathematical instruments, and then use them to discuss general expressions for the distribution of the walker's displacement after a given number of steps in one dimension and in higher dimensions. It moreover discusses moments of displacement, provided these moments exist. The chapter then considers a simple approach to the central limit theorem, and discusses situations, when this breaks down (corresponding to the cases when the second moment of step lengths diverges).