Result: An almost sure central limit theorem with generalized moments for irrational rotations

Let \(\{S_n\), \(n\geq 1\}\) be the partial sums of a sequence \(\{X_n\}\) of i.i.d. random variables with \(EX_1= 0\), \(EX_1^2= 1\). \textit{M. T. Lacey} and \textit{W. Philipp} [Stat. Probab. Lett. 9, No. 3, 201-205 (1990; Zbl 0691.60016)] proved that, with probability one, \[ \lim_{N\to \infty} (\log N)^{-1} \sum_{j=1}^N j^{-1} \delta_{\{S_j/ j^{1/2}\}}= N(0,1)\text{ in distribution}, \] where \(\delta_z\) is the Dirac measure at \(z\). This almost sure central limit theorem is extended to the case of irrational rotations of a torus. Sufficient conditions are presented which guarantee the existence of a function \(g\) such that the Lacey-Philipp result holds with \(\delta_{\{S_j/ j^{1/2}\}}\) replaced by \(\delta_{\{S_j(g)/ \sigma_j\}}\) for a real sequence \(\{\sigma_n\}\) such that \(0< \inf_{n\geq 1} n^{-1/2} \sigma_n\leq \sup_{n\geq 1} n^{-1/2} \sigma_n< \infty\). Rates of convergence are also established.