Asymptotic laws for compositions derived from transformed subordinators

By Alexander V. Gnedin, Jim Pitman, Marc Yor. 2004

57 citations recorded by the source corpus.

A random composition of n appears when the points of a random closed set R ⊂ [0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K n of this composition and other related functionals under the assumption that R = Φ(S • ), where (S t , t ≥ 0) is a subordinator and Φ:[0,∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of K n when the Levy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function Φ(x) = 1 - e -x , we establish a connection between the asymptotics of K n and the exponential functional of the subordinator.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3164cb307af335436265857a1c54c60b153d137c

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In Applied Mathematics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Combinatorial Stochastic Processes (2006, background): “13, it has to be seen that the α-diversity limn |Πn|/n of such a partition equals T−α almost surely, but this can be seen as in [171].”
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗ (2007, background): “See the recent work on composition structures [20, 21, 3] for more in this direction.”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, background): “ As indicated above, the local time process (L(u),0 ≤ u ≤ 1) can be described directly in terms of ξ or Zα,θ: in the setting of Proposition 6, we have L(1−exp(−ξt))=Γ(1−α) Z t 0 (13) exp(−αξs)ds; cf. [14], Section 5. The right-continuous inverse of L satisﬁes L−1(ℓ)=1−exp(−ξT ℓ ) where Tℓ =Γ(1−α) Z ”
Limit theorems for the number of occupied boxes in the Bernoulli sieve (2010, background): “1 in [15] and Corollary 21 in [8] (interesting extensions can be found in [24]).”