Regenerative composition structures

By Alexander V. Gnedin, Jim Pitman. 2003

131 citations recorded by the source corpus.

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative

Subfields: Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/25cb51161950a7793f863a5dac842becc32a4416

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Citations in the corpus

Asymptotic laws for compositions derived from transformed subordinators (2004, methodology): “For compositions belonging to the two-parameter family [16, 21, 25], the proper format for Kn is n α for parameters (α, θ) with 0< α< 1.”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, background): “While the birth order is well-known to be size-biased, we show that the compositions in spinal order form a regenerative composition structure in the sense of Gnedin and Pitman [13], which is weakly sampling consistent for all 0 ≤ α ≤ 1 and θ ≥ 0, but not strongly so unless θ = α (Proposition 6 (i) ”
Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams (2005, methodology): “Similarly, the realisation of - by (10) is a version of the representation of extreme exchangeable ordered partitions, also called composition structures [13, 17].”
Combinatorial Stochastic Processes (2006, unknown): “See for instance [362, 170].”
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗ (2007, background): “This theme lies outside the scope of this paper, and we only mention one generalization of (53) that appears in the theory of exchangeable ordered partitions [18, 21].”
Limit theorems for the number of occupied boxes in the Bernoulli sieve (2010, background): “In this case (see, for instance, Example 2 in [14] or Section 5.”
Exchangeable Gibbs partitions and Stirling triangles (2004, methodology): “According to a Kingman’s paintbox representation [22, 9, 10 ], every such exchangeable partition has the same distribution as � constructed from some random closed set Z ⊂ [0,1], as follows: let u1, u2, . . . be independent uniform [0,1] variables, independent of Z, and let distinct integers i and ”
The Bernoulli sieve (2004, unknown): “This conclusion is in good accord with a general point that the composition Cn is a proper combinatorial analogue of the regenerative set R, as in Gnedin (2003), Gnedin and Pitman (2003) and Gnedin et al. (2003).”
Bayesian Nonparametrics: Models beyond the Dirichlet process (2009, unknown): “An interesting and closely related class of random probability measures is given by Gibbs–type priors, introduced in Gnedin and Pitman (2005a).”