Distribution Functions for Subsequences of Generalized Van Der Corput Sequences

For an integer \( b > 1 \) let \( (\phi_b(n))_{n \geq 0} \) denote the van der Corput sequence base \( b \) in \( [0,1) \). Answering a question of O. Strauch, C. Aistleitner and M. Hofer showed that the distribution function of

\[
\big(\phi_b(n), \phi_b(n+1), \ldots, \phi_b(n+s-1)\big)_{n \geq 0} \quad \text{on} \quad [0,1)^s
\]

exists and is a copula. The first and third authors of the present paper showed that this phenomenon extends to a broad class of subsequences of the van der Corput sequence. In this result we extend this paper still further and show that this phenomenon is also true for more general numeration systems based on the beta expansion of W. Parry and A. Rényi.