The central limit theorem for Rademacher system (Q1344804)

Let \(\{\tau_ i\}\) denote a Rademacher system of random variables defined by \[ \tau_ 1(\omega):= -\text{\textbf{1}}_{[0,1/2)} (\omega)+ \text{\textbf{1}}_{[1/2,1)}(\omega)\qquad\text{and}\qquad \tau_ i(\omega):= \tau_ 1(2^{i- 1}\omega),\;i\geq 2. \] Put \(X^{(m,\alpha)}_ n(\omega) := X^{(m)}(\omega+ n\alpha)\) for \(n\in Z\) and \(\alpha\in R\), where \(X^{(m)}(\omega):= {1\over \sqrt m} \sum^ m_{i= 1} \tau_ i(\omega)\). The author proves that for almost all \(\alpha\) any finite-dimensional distribution of \(\{X^{(m,\alpha)}_ n\}_{n\in Z}\) converges weakly to the multidimensional standard normal law as \(m\to \infty\). A rate of convergence of \(R^{(m, \alpha)}(k):= E(X^{(m, \alpha)}_ n, X^{(m, \alpha)}_{n+ k})\) for \(m\to \infty\) is also established.