A key problem in optimal input design is that the solution depends on system parameters to be identified. In this contribution we provide formal results for convergence and asymptotic optimality of an adaptive input design method based on the certainty equivalence principle, i.e. for each time step an optimal input design problem is solved using the present parameter estimate and one sample of this input is applied to the system. The results apply to stable ARX systems with the input restricted to be generated by white noise filtered through an FIR filter, or a binary signal obtained from the latter by a static nonlinearity.