Distribution functions for subsequences of the van der Corput sequence

For an integer b>1$b>1$ let _(?b(n))n≥0${\left({?}_b\left(n\right)\right)}_{n\ge 0}$ denote the base b$b$ van der Corput sequence in [0,1)$[0,1)$. Answering a question of O. Strauch, C. Aisleitner and M. Hofer showed that the distribution function of _{(?b(n),?b(n+1),…,?b(n+s−1))n≥0}${({?}_b\left(n\right),{?}_b(n+1),\dots ,{?}_b(n+s-1))}_{n\ge 0}$ on [0,1)^s${[0,1)}^s$ exists and is a copula. In this note we show that this phenomenon extends to a broad class of subsequences of the van der Corput sequences.