Distribution functions for subsequences of the van der Corput sequence

For an integer b>1b>1 let (ϕb(n))n≥0(ϕb(n))n≥0 denote the base bb 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 (ϕb(n),ϕb(n+1),…,ϕb(n+s−1))n≥0(ϕb(n),ϕb(n+1),…,ϕb(n+s−1))n≥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.