A central limit theorem for stationary \(\rho\)-mixing sequences with infinite variance (Q1102025)

A central limit theorem is proved for strictly stationary sequences \(X_ 1,X_ 2,..\). which are weakly dependent in the sense of \(\rho\)-mixing (maximal correlation). The \(\rho\)-mixing coefficients \(\rho\) (n) are assumed to satisfy \(\rho (1)<1\) and \(\rho (2^ k)\) summable (which is essentially the slowest possible rate at which \(\rho\) (n) must converge to zero in order to obtain the result). If the marginal distribution of \(X_ 1\) satisfies the classical tail conditions [see \textit{I. A. Ibragimov} and \textit{Y. V. Linnik}, Independent and stationary sequences of random variables (1971; Zbl 0219.60027), p. 83], then the partial sums are attracted to a normal law.