On nested infinite occupancy scheme in random environment

Abstract

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.

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

“…, K (u) n ( ) u≥0 for each ∈ N, properly normalized and centered, as n → ∞. In particular, according to Corollary 4.3 in [15] the following multivariate central limit theorem holds in the GEM(0, 1) case: for each ∈ N, as n → ∞,…”

“…This means that (1.1) holds with independent W k 's having a uniform distribution on [0, 1], that is, P{W k ∈ dx} = 1 (0,1) (x)dx. In [15], for a rather wide class of exponentially decaying random probabilities (environments), a functional limit theorem was proved for the random process K (u) n (1), . .…”

“…Following [5] and [15] (see also [22,7]) we now define a nested infinite occupancy scheme in random environment. This means that we construct a nested sequence of the environments (P k ) and the corresponding 'boxes' so that the same collection of 'balls' is thrown into all 'boxes'.…”

“…Bertoin in Theorem 1 of [5] proved that K n (j n )/f (n) converges almost surely to a limit random variable which includes as a factor the terminal value of the Biggins martingale (it is the presence of the Biggins martingale limit which manifests the influence of the boundary). Here, f (x) is a regularly varying function which is given explicitly, and (j n ) n∈N is the sequence of integers satisfying lim n→∞ (j n − a log n) = b for some a > 0 and b ∈ R. In [15] the behavior near the root of a weighted branching process tree was investigated. More precisely, the occupancy of any EJP 25 (2020), paper 123. fixed number of fixed (not depending on n) generations was analyzed.…”

Weak convergence of random processes with immigration at random times

“…With the help of a branching property, we obtain the basic decomposition where for are independent copies of which are also independent of the T ( v ), . Motivated by an application to certain nested infinite occupancy schemes in a random environment, the authors of the recent article [4] proved functional limit theorems in for with appropriate centering and normalizing functions and . The standing assumption of [4] is that the positions are given by , where , are positive random variables with an arbitrary joint distribution satisfying a.s.…”