On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I

TL;DR

This paper studies the occupancy of intermediate generations in a nested scheme generated by stick-breaking, showing convergence of normalized occupied box counts to a Brownian motion functional, with analysis of a general stick-breaking case.

Contribution

It introduces a detailed analysis of occupancy in intermediate generations of a nested stick-breaking scheme, revealing weak convergence to a Brownian motion functional.

Findings

Normalized occupied box counts converge to a Brownian motion functional.

The results apply to uniform and more general stick-breaking distributions.

Intermediate generation occupancy exhibits a universal asymptotic behavior.

Abstract

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 scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment. Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_n u\rfloor$ for $u>0$, where $j_n$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_n(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.