Limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is a version of the classical ‘balls-in-boxes’ occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. A refined approach based on the standard renewal theory allows us to remove a moment constraint and to cover the cases left open in previous studies.