Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

Abstract: The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}<sup>{*}(t)$ of boxes containing at most $[n<sup>{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}<sup>{<em>}(t)/K_{n}^{</em>}(1)$, as $n\to\infty$. The limit processes for $K_{n}<sup>{*}(t)$ are of the form $(X(1)-X((1-t)-))<em>{t\in[0,1]}$, where $X$ is either a Brownian motion, a spectrally negative stable L\'evy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for $K</em>{n}<sup>{<em>}(t)/K_{n}^{</em>}(1)$ is a L\'evy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of $K_{n}<sup>{*}(1)$. First, for any Karlin occupancy scheme with deterministic probabilities $(p_{k})<em>{k\ge 1}$, we obtain an approximation, uniformly in $t\in[0,1]$, of the number of boxes with at most $[n<sup>{t}]$ balls by a counting function defined in terms of $(p</em>{k})_{k\ge 1}$. Second, we prove several FLTs for the number of visits to the interval $[0,nt]$ by a perturbed random walk, as $n\to\infty$.