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.
Highlights
The infinite occupancy scheme is obtained by allocating ‘balls’ independently over an infinite collection of ‘boxes’ 1, 2, . . . with probability pr of hitting box r, r ∈ N, whereNested occupancy scheme in random environment r≥1 pr = 1
The infinite occupancy scheme in a random environment has received much less attention. The definition of the latter schemes assumes that probabilities (Pr)r∈N are random with an arbitrary joint distribution satisfying r≥1 Pr = 1 a.s., and that conditionally given (Pr)r∈N ‘balls’ are allocated independently with probability Pr of hitting box r, r ∈ N
One may expect that to the known phenomenon for the infinite occupancy in random environment the regular variation of the counting function of probabilities entails the a.s. convergence of the number of occupied boxes, properly normalized, whereas the slow variation entails distributional fluctuations of the number of occupied boxes, properly normalized and centered
Summary
The infinite occupancy scheme is obtained by allocating ‘balls’ independently over an infinite collection of ‘boxes’ 1, 2, . . . with probability pr of hitting box r, r ∈ N, where. The nested occupancy schemes in random environment are interesting because these include features of the infinite occupancy schemes and the weighted branching processes Even though the latter two objects are rather popular, they belong to distinct areas of probability theory. One may expect that to the known phenomenon for the infinite occupancy in random environment (that is, when attention is restricted to the first generation) the regular variation of the counting function of probabilities entails the a.s. convergence of the number of occupied boxes, properly normalized, whereas the slow variation entails distributional fluctuations of the number of occupied boxes, properly normalized and centered
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