Functional Central Limit Theorem for Empirical Processes

Here we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ‎-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α‎-dependent sequences whose α‎-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β‎-dependent sequences are also investigated by directly proving tightness of the empirical process.

Stationary Sequences in a Random Time Scenery

In this chapter, we analyze the asymptotic behavior of the partial sums process associated with examples of stationary sequences in a random time scenery. The examples considered are stationary sequences sampled by shifted renewal Markov chains and random walks in a strictly stationary scenery. The asymptotic behavior of the partial sums process is essentially investigated with the help of the weak invariance principles stated in Chapter 4. More precisely, for the partial sums process associated with a stationary process sampled by a renewal Markov chain stated at zero, due to the non-stationarity of the underlying sequence, the functional CLT is obtained as an application of the functional CLT for non-stationary sequences developed in Section 4.4. In the case where we are sampling a strictly stationary random scenery by a random walk, stationarity is preserved, and the invariance principle is then derived by using the functional CLT under Maxwell–Woodroofe condition.

Gaussian Approximation under Asymptotic Negative Dependence

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.

Reversible Markov Chains

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances between the variables. We present here the traditional form of the martingale approximation including forward and backward martingale approximations. Special attention is given to maximal inequalities which are building blocks for the functional limit theorems. When the covariances are summable we present the functional central limit theorem under the standard normalization √n. When the variance of the partial sums are regularly varying with n, we present the functional CLT using as normalization the standard deviation of partial sums. Applications are given to the Metropolis–Hastings algorithm.