Short Range and Long Range Dependence

In this section a discussion of the evolution of a notion of strong mixing as a measure of short range dependence and with additional restrictions a sufficient condition for a central limit theorem, is given. In the next section I will give a characterization of strong mixing for stationary Gaussian sequences. In Sect. 3 I will give a discussion of processes subordinated to Gaussian processes and in Sect. 4 results concerning the finite Fourier transform is noted. In Sect. 5 a number of open questions are considered. In an effort to obtain a central limit theorem for a dependent sequence of random variables in [12], I made use of a blocking argument of S.N. Bernstein [1] and was led to what I called a strong mixing condition [2, 12]. In the blocking argument big blocks are separated by small blocks. Consider a sequence of random variables Xn, n D : : : ; 1; 0; 1; : : :. Let Bn and Fm be the -fields generated by Xj, j n and Xj, j m, respectively. If