This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.
We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.
The paper concerns the classical occupancy scheme with infinitely many boxes. We establish approximations to the distributions of the number of occupied boxes, and of the number of boxes containing exactly r balls, within the family of translated Poisson distributions. These are shown to be of ideal asymptotic order, with respect both to total variation distance and to the approximation of point probabilities. The proof is probabilistic, making use of a translated Poisson approximation theorem of Röllin (2005).
A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assum…
Consider a weighted branching process generated by a point process on $[0,1]$, whose atoms sum up to one. Then the weights of all individuals in any given generation sum up to one, as well. We define a nested occupancy scheme in random environment as the sequence of balls-in-boxes schemes (with random probabilities) in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, and each ball starts at the root and…
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 random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.
The paper is concerned with the classical occupancy scheme with infinitely many boxes, in which $n$ balls are thrown independently into boxes $1,2,...$, with probability $p_j$ of hitting the box $j$, where $p_1\geq p_2\geq...>0$ and $\sum_{j=1}^\infty p_j=1$. We establish joint normal approximation as $n\to\infty$ for the numbers of boxes containing $r_1,r_2,...,r_m$ balls, standardized in the natural way, assuming only that the variances of these counts all tend to infinity. The proof of this approximation is based on a de-Poissonization lemma. We then review sufficient conditions for the var…
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1...W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ are independent copies of a random variable $W$ taking values in $(0,1)$. Assuming that the number of balls equals $n$, let $L_n$ denote the number of empty boxes within the occupancy range. The paper proves that, under a regular variation assumption, $L_n$, properly normalized without centering, weakly converges to a functional of an inverse stable subordinator. Proofs rely upon the observation that $(\log P_k)$ is a perturbed random walk. In partic…
Let $(p_k)_{k\in\mathbb{N}}$ be a discrete probability distribution for which the counting function $x\mapsto \#\{k\in\mathbb{N}: p_k\geq 1/x\}$ belongs to the de Haan class $Π$. Consider a deterministic weighted branching process generated by $(p_k)_{k\in\mathbb{N}}$. A nested Karlin's occupancy scheme is the sequence of Karlin balls-in-boxes schemes in which boxes of the $j$th level, $j=1,2,\ldots$ are identified with the $j$th generation individuals and the hitting probabilities of boxes are identified with the corresponding weights. The collection of balls is the same for all generations, …
In the classical occupancy scheme, one considers a fixed discrete probability measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\bf p}(k) : k\in\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chains
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.
The Bernoulli sieve is an infinite occupancy scheme obtained by allocating the points of a uniform $[0,1]$ sample over an infinite collection of intervals made up by successive positions of a multiplicative random walk independent of the uniform sample. We prove a law of the iterated logarithm for the number of non-empty (occupied) intervals as the size of the uniform sample becomes large.
A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer $j$, identify the $j$th generation individuals with the $j$th generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the fol…
Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a `box'. Given the weights `balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy …
The Bernoulli sieve is the infinite "balls-in-boxes" occupancy scheme with random frequencies $P_k=W_1... W_{k-1}(1-W_k)$, where $(W_k)_{k\in\mn}$ are independent copies of a random variable $W$ taking values in $(0,1)$. Assuming that the number of balls equals $n$, let $L_n$ denote the number of empty boxes within the occupancy range. In the paper we investigate convergence in distribution of $L_n$ in the two cases which remained open after the previous studies. In particular, provided that $\me |\log W|=\me |\log (1-W)|=\infty$ and that the law of $W$ assigns comparable masses to the neighbo…
The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to infinity, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random wal…
We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance V_n tends to infinity. In this work, we mainly focus on the opposite case where V_n is bounded, and derive a simple necessary and sufficient condition for convergence of V_n to a finite limit, thus settling a long-standing question raised by Karlin in the seminal paper of 1967. One striking consequence of our result is that the possible limit may on…
Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme
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 $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\to\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\in[0,1]}$, where $X$ is either a Brownian motion, a spec…
We consider an occupancy scheme in which "balls" are identified with $n$ points sampled from the standard exponential distribution, while the role of "boxes" is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behavior of five quantities: the index $K_n^*$ of the last occupied box, the number $K_n$ of occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most $K_n^*$, the index $W_n$ of the first empty box and the number of balls $Z_n$ in the last occupied box. It is shown that the limiting distribution o…
Let $μ$ be a probability measure (or corresponding random variable) such that all moments $μ_n$ exist. Knowledge of the moments is not sufficient to determine infinite divisibility of the measure; we show also that infinitely divisible, and in particular lognormal, distributions lose infinitely divisibilty when censored in certain ways even if all moments are arbitrarily close to those of the uncensored distribution. The moments of a composition of k copies of $μ$ are expressed as combinatorial compositions of the $μ_n$. We express the moments of the compositions in the context of occupancy pr…
