Input–output properties of the Page–Hinkley detector

We consider the stochastic input–output properties of a simple non-linear dynamical system, the so-called Page–Hinkley detector, playing a key role in change detection, and also in queuing theory. We show that for LL-mixing inputs with negative expectation the output process of this system is LL-mixing. The result is applied to get an upper bound for the false alarm rate. The proof is then adapted to get a similar result for the case of random i.i.d. inputs. Possible extensions and open problems are given in the discussion.