  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. Refining the
approach based on the standard renewal theory we remove a moment constraint to
cover the cases left open in previous studies.
