Asymptotic equidistribution of congruence classes with respect to the convolution iterates of a probability vector

Consider a positive integer d and a positive probability vector f over the numbers 0,…,ℓ. The n-fold convolution f∗n of f is a probability vector over the numbers 0,…,nℓ, and these can be partitioned into congruence classes modulo d. The main result of this paper is that, asymptotically in n, these d congruence classes have equiprobability 1/d. In the motivating application, one has N containers of capacity d and repeatedly retrieves one item from each of M randomly selected containers (0<M<N); containers are replenished to full capacity when emptied. The result implies that, over the long term, the number of containers requiring replenishment is M/d. This finding is relevant wherever one would be interested in the steady-state pace of replenishing fixed-capacity containers.