On weak convergence of sampled dynamical systems (Q2756233)

The following generalization of a result of \textit{M. T. Lacey} [J. Anal. Math. 61, 47-59 (1993; Zbl 0790.60027)] is announced. Let \(B_H\) denote the fractional Brownian motion with index \(0< H< 1\), \(Rf= \sum_{k\in\mathbb{N}} p(k)T^k_\alpha f\), where \(p\) is a probability vector and \(T_\alpha\) is the rotation by the angle \(\alpha\). If \(\gamma\) denotes the Diophantine type of \(\alpha\), then for \(a<(1-H)/\gamma\) there exists a function \(f\in \text{Lip}(\alpha)\) such that \({1\over n^H} \sum^{[nt]-1}_{k=0} f\circ R^k\Rightarrow B_H(t)\) in finite-dimensional distributions. If \(a> (1-H)/\gamma\), such a function cannot exist. The result is applied to random partial sums and the corresponding convergence to \(B_H\).