Self-normalized central limit theorem for sums of weakly dependent random variables (Q1322910)

Let \(\{x_ n, n\geq 1\}\) be a strictly stationary sequence of weakly dependent r.v.s with finite mean and variance, \(\text{var}(S_ n/n)\to \sigma^ 2>0\) and central limit theorem. The authors present two estimators of \(\sigma^ 2\), \(B_{1n}=(\log n)^{-1}\sum^ n_ 1 i^{-1/2}|\overline X_ i-\overline X_ n|\) and \(B^ 2_{2n} =(\log n)^{-1}\sum^ n_ 1(\overline X_ i-\overline X_ n)^ 2\), where \(\overline X_ i\) is the sample mean based on first \(i\) observations. The authors prove their \(L_ 2\), \(L_ 1\) and strong consistency as well as their rate of convergence for strong mixing, \(P\)- mixing and associated sequences. Assuming strong approximation for stationary weakly dependent sequences they also obtain asymptotic normality and iterated logarithm type results as auxiliary results.