Ranked Fragmentations

By J. Berestycki. 2006

50 citations recorded by the source corpus.

In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time behavior of a ranked fragmentation.

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6a2248f476c03465cc817577da9f12aa4f9ec2e4

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In Management Science and Operations Research

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Self-similar fragmentations derived from the stable tree II: splitting at nodes (2004, background): “By [5], the laws of the self-similar fragmentations are characterized by a 3-tuple (β, c, ν), where β is the self-similarity index, c ≥ 0 is an erosion coefficient and, more importantly, ν is a σ -finite dislocation measure on S that integrates the map s → 1 − s1.”
Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures (2004, background): “For an element π of P the restriction of π to [n] is π|[n] and we identify each π ∈ P with the sequence (π|[1], π|[2], .”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, background): “For some a ∈ R, called the self-similarity index, it holds that conditionally given X(t) = (s1, s2, . . .), the process (X(t + t ′), t ′ ≥ 0) has same distribution as the process whose value at time t ′ is the decreasing rearrangement of the sequences siX(i)(sai t ′), i ≥ 1, where (X(i), i ≥ 1) are ”
Discretization Methods for Homogeneous Fragmentations (2004, background): “More precisely, (1) is the necessary and sufficient condition for a measure ν on S to be the dislocation measure of some homogeneous fragmentation (see [3] and [4]).”
The Genealogy of Self-similar Fragmentations with Negative Index as a Continuum Random Tree (2004, unknown): “By a result of Bertoin [7] and Berestycki [4], the laws of such fragmentation processes are characterized by a 3-tuple ( α, c, ν ), where α is the index, c ≥ 0 is an “erosion” constant, and ν is a σ -ﬁnite measure on S that integrates s 7→ 1 − s 1 such that ν ( { (1 , 0 , 0 . . . ) } ) = 0.”
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006, background): “We notice that if (Π(t), t ≥ 0) is a self-similar P-valued fragmentation, then (|Π(t)|↓, t ≥ 0) is a self-similar fragmentation with values in S↓, as defined in the introduction (and any S↓-valued fragmentation can be represented in this form, see [6, 8]).”
Combinatorial Stochastic Processes (2006, background): “See [357], [141], [35] and papers of Bertoin cited above, for treatment of various technical problems related to regularity of different partition-valued Markov processes, transformations between different representations, and so on.”
Multifractal spectra and precise rates of decay in homogeneous fragmentations (2006, background): “…0 conditionally on the open set F ( t ) = ∪ i ∈ I J i ( t ), the interval fragmentation F ( t + s ) has the same law as F 1 ( s ) ∪ F 2 ( s ) ∪ ... where for each i , F i ( s ) is a subset of J i ( t ) and has the same distribution as the image of F ( s ) by the homothetic map (0 , 1) → J i ( t ).”