A phase transition for the heights of a fragmentation tree

Joseph, Adrien

doi:10.1002/rsa.20340

Cited by 10 publications

(19 citation statements)

References 27 publications

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“…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'.…”

Section: On Intermediate Levels Of Nested Occupancy Scheme In Random Environment Generated By Stick-breaking I 1 Introductionmentioning

confidence: 99%

“…For j ∈ N and t > 0, denote by ρ j (t) := #{|u| = j : P (u) ≥ 1/t} the counting function for the probabilities in the jth generation. The earlier investigations of the occupancy schemes in random environment by J. Bertoin [5], A. Joseph [22] and S. Businger [7]) focused on the behavior near the boundary of a weighted branching process tree. This setting enabled the authors to make use of various asymptotic properties of the weighted branching process such as the convergence of the Biggins martingale and large deviations.…”

Section: On Intermediate Levels Of Nested Occupancy Scheme In Random Environment Generated By Stick-breaking I 1 Introductionmentioning

confidence: 99%

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

Buraczewski

Dovgay²,

Iksanov³

2020

Electron. J. Probab.

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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 jth 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 jnu for u > 0, where jn diverges to infinity at a sublogarithmic rate as n becomes large. Denote by Kn(j) the number of occupied (ever hit) boxes in the jth generation. It is shown that the finite-dimensional distributions of the process (Kn( jnu ))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.

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Section: On Intermediate Levels Of Nested Occupancy Scheme In Random Environment Generated By Stick-breaking I 1 Introductionmentioning

confidence: 99%

Section: On Intermediate Levels Of Nested Occupancy Scheme In Random Environment Generated By Stick-breaking I 1 Introductionmentioning

confidence: 99%

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

Buraczewski

Dovgay²,

Iksanov³

2020

Electron. J. Probab.

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“…There is a version of the nested Karlin's occupancy scheme, called nested occupancy scheme in random environment, in which the distribution (p k ) k∈N is random. Such a model was introduced in [2] and further investigated in [6], [7], [15], [18], [19]. In [2] and [19] the asymptotics of the number of occupied boxes K (j) n and related quantities was analyzed at the generations j of order log n. Some results of the last two cited papers apply to the nested Karlin occupancy scheme.…”

Section: Relevant Literaturementioning

confidence: 99%

“…Such a model was introduced in [2] and further investigated in [6], [7], [15], [18], [19]. In [2] and [19] the asymptotics of the number of occupied boxes K (j) n and related quantities was analyzed at the generations j of order log n. Some results of the last two cited papers apply to the nested Karlin occupancy scheme. We are not aware of any articles which would treat the generations j with j = j n → ∞ and j n = o(log n) as n → ∞ of the nested Karlin occupancy scheme.…”

Section: Relevant Literaturementioning

confidence: 99%

A functional limit theorem for nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

Iksanov¹,

Kabluchko²,

Kotelnikova³

2021

Preprint

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Let (p k ) k∈N be a discrete probability distribution for which the counting function x → #{k ∈ N : p k ≥ 1/x} belongs to the de Haan class Π. Consider a deterministic weighted branching process generated by (p k ) k∈N . A nested Karlin's occupancy scheme is the sequence of Karlin balls-in-boxes schemes in which boxes of the jth level, j = 1, 2, . . . are identified with the jth 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 moves along the tree of the deterministic weighted branching process according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights.Assuming there are n balls, denote by K n (j) the number of occupied (ever hit) boxes in the jth level. For each j ∈ N, we prove a functional limit theorem for the vector-valued process (K⌊e T +u ⌋ ) u∈R , properly normalized and centered, as T → ∞. The limit is a vector-valued process whose components are independent stationary Gaussian processes. An integral representation of the limit process is obtained.

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“…Following [9,12,30] we shall study a nested infinite occupancy scheme in random environment. In this context we regard (P k ) k∈N as a random fragmentation law (with P k > 0 and k∈N P k = 1 a.s.).…”

Section: Introductionmentioning

confidence: 99%

On nested infinite occupancy scheme in random environment

Gnedin

Iksanov

2020

Probab. Theory Relat. Fields

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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.

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A phase transition for the heights of a fragmentation tree

Cited by 10 publications

References 27 publications

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

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

A functional limit theorem for nested Karlin's occupancy scheme generated by discrete Weibull-like distributions

On nested infinite occupancy scheme in random environment

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