Examples and Counterexamples

AbstractIn this chapter we further comment on the sharpness of several results presented in this monograph, by presenting examples and counterexamples. We study first the moment properties of the renewal Markov chain introduced in Chapter 11. This allows us to show that the Maxwell–Woodroofe projective condition introduced in Chapter 4 is essentially optimal for the partial sums of a stationary sequence in L2 to satisfy the central limit theorem under the standard normalization √n. Moreover, we also investigate the sharpness of the Burkholder-type inequality developed in Chapter 3 via Maxwell–Woodroofe-type characteristics. In the last part of this chapter, we analyze several telescopic-type examples allowing us to elucidate the fact that a CLT behavior does not imply its functional form under any normalization. Even in the case when the variance of the partial sums is linear in n, the CLT does not necessarily imply the invariance principle.