Stability of a class of non-uniform random number generators (Q1323188)

Let \((z_ n)\) be a sequence of nonzero reals satisfying the recurrence \(z_{n+1}= 2- b_ n z_ n^{-1}\), where \((b_ n)\) is a sequence of reals with \(b_ n= 1+d+ O(n^{-\lambda})\) and \(\sum_{m=n}^ \infty | b_{m+1}- b_ m|= O(n^{-\lambda})\) and \(d>0\) and \(\lambda>1\) are constants. Then it is proved that \((z_ n)\) has an asymptotic distribution function mod 1 given by \[ F(z)= z+ {1\over\pi} \arctan {{\sin 2\pi z} \over {\exp (2\pi \sqrt{d})- \cos 2\pi z}}. \] An upper bound for the discrepancy of \((z_ n)\) with respect to the distribution function \(F\) is established under an additional arithmetic condition on \(d\). As an interesting consequence it is noted that slight perturbations of the \(b_ n\) do not affect the distribution function \(F\).