A direct proof of dependence vanishing theorem for sequences generated by Weyl transfor\-mation (Q1890295)

Let \(b\geq 2\) be an integer and \(d^{(m)}(x)\) be the \(m\)th digit of \(x\geq 0\) in the decimal part of its base \(b\) expansion. Let \(X_l^{(m)}\) be \(\{0,\ldots,b-1\}\)-valued function on \([0,1)^2\) such that \[ X_l^{(m)}(x,\alpha)=\sum_{k=1}^m d^{(k)}(x+l\alpha) \bmod b. \] Assume that \(P\) is a measure on \([0,1)\) such that \(\{d^{(i)}\}_i\) is independent with respect to it and \[ \liminf_i \min_{0\leq c<b} P(d^{(i)}=c)>0. \] The author shows that for any \(\alpha\) with base \(b\) expansion containing any finite sequence infinitely many times, the process \(\{X_n^{(m)}(\cdot,\alpha)\}_{n=0}^\infty\) on \(([0,1),P)\) converges in law to \(\{0,\ldots,b-1\}\)-valued fair i.i.d.\ when \(m\rightarrow \infty\). In case \(b=2\) and \(P\) is the Lebesgue measure this result was proven by \textit{H. Sugita} [Monte Carlo Methods Appl. 1, No.1, 35--57 (1995; Zbl 0827.65002)].