Short-Range Dependence

Abstract

This chapter introduces some simple ideas. We investigate conditions on time series such that the standard limit theorems obtained for independent identically distributed sequences still hold. After a general introduction to weak-dependence conditions an example states the fact that the most classical strong-mixing condition from (Rosenblatt (1956). Proc Natl Acad Sci U S A 42:43–47.) Rosenblatt (1956) may fail to work, see (Andrews (1984). J Appl Probab 21:930–934.) Andrews (1984). When dealing with any weak-dependence condition (including strong mixing), additional decay rates and moment conditions necessary to ensure CLTs. Decay rates will be essential to derive asymptotic results. Coupling arguments as proposed in Sect. 7.4.2 are widely used for this. Finally to make clearer the need for decay rates, we explain how CLTs may be proved under such assumptions. The monograph (Dedecker, Doukhan, Lang, León, Louhichi, Prieur (2007). Lect Notes Stat 190.) (Dedecker et al. 2007) is used as the reference for weak-dependence; in this monograph we developed more formal results together with their proofs. We refer a reader to this monograph for more rigorous results.