Logarithmic Combinatorial Structures: A Probabilistic Approach

By Richard Arratia, A. Barbour, S. Tavaré. 2003

610 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6c4bda4f818761a4479c95e332f3aa0c5ad3c299

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Citations in the corpus

Asymptotic laws for compositions derived from transformed subordinators (2004, neutral): “ar X iv :m at h/ 04 03 43 8v 3 [ m at h. PR ] 2 4 M ay 2 00 6 The Annals of Probability 2006, Vol. 34, No. 2, 468–492 DOI: 10.1214/009117905000000639 c© Institute of Mathematical Statistics, 2006”
Random Graphs and Complex Networks (2016, neutral): “[16].”
Limit theorems for the number of occupied boxes in the Bernoulli sieve (2010, background): “For instance, choosing fn(t) := 1[1−1/n, 1+1/n)(t) + 2 · 1[1+1/n, 2](t) and f(t) := 2 · 1[1, 2](t), the fn do converge to f in the M1-topology, but not in the J1-topology on D[0, 2].”
Recent progress in coalescent theory (2009, neutral): “49) in [9], which credits Billingsley [33] for this result, and [62] for a different proof using size-biased ordering.”
Exchangeable Gibbs partitions and Stirling triangles (2004, background): “For the Stirling numbers of the first kind there is an asymptotic formula [3]”
Size bias for one and all (2013, background): “The infinite divisibility of geometric and negative binomial distributions plays a key role in estimates comparing logarithmic combinatorial structures with their limits; see [7].”
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (2011, methodology): “76]; our results apply only to random objects with a given number of components (in some cases), but similar methods are useful for the general case; see Kolchin [76] and Arratia, Barbour and Tavar´e [7]. 12. Preliminaries Proof of Lemma 3.1. (i): Since Φ′(t) = P∞ k=0 kwkt k−1 has the same radius of”
The component sizes of a critical random graph with given degree sequence (2010, background): “It is well known that the distribution of cycle lengths is given by the Ewens’s sampling formula ESF(1/2), and thus the size of the largest component divided by n converges in distribution to a nondegenerate distribution on [0,1]; see [3], Lemma 5.”
The Bernoulli sieve (2004, positive): “This structure is well understood; see Arratia et al. (2002) for a recent account and Pitman (2002), Gnedin (2003) and Gnedin and Pitman (2003) for generalizations.”
BETA-COALESCENTS AND CONTINUOUS STABLE RANDOM TREES (2006, background): “Among many others, we refer the reader to [4] and to [28] for different proofs of (8).”
Random Recursive Trees and the Bolthausen-Sznitman Coalesent (2005, neutral): “718”
Fundamentals of Stein's method (2011, background): “We could also handle the number of i-cycles in a random permutation, but the analysis is a bit more tedious and is not worth pursuing due to the fact that so much more is known in this example - see [2] for a thorough account.”
Mixing times for random k-cycles and coalescence-fragmentation chains (2010, background): “Whenever k is even, and thus μ is uniform on Sn, (29) is a classical result of Diaconis–Pitman and of Barbour, with explicit upper bound of 4K/n (see [4] or the discussions around [3], Theorem 2, and [2], Theorem 4.”