We derive necessary and sufficient conditions for a sum of i.i.d.random variables $\sum_{i=1}^n X_i/b_n$ –where $\frac {b_n} n \downarrow 0$ ,but $\frac {b_n} {\sqrt n} \uparrow \infty$ – to satisfy a moderate deviationsprinciple. Moreover we show that this equivalence is a typical moderatedeviations phenomenon. It is not true in a large deviations regime.

We investigate the number of iterations needed by an addition algorithm due to Burks et al. if the input is random. Several authors have obtained results on the average case behaviour, mainly using analytic techniques based on generating functions. Here we take a more probabilistic view which leads to a limit theorem for the distribution of the random number of steps required by the algorithm and also helps to explain the limiting logarithmic periodicity as a simple discretization phenomenon.