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Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also de ..."
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
Cycles and unicyclic components in random graphs
 Combin. Probab. Comput
"... Abstract. The sizes of the cycles and unicyclic components in the random graph G(n, n/2 ± s), where n 2/3 ≪ s ≪ n, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given; these distr ..."
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Cited by 9 (4 self)
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Abstract. The sizes of the cycles and unicyclic components in the random graph G(n, n/2 ± s), where n 2/3 ≪ s ≪ n, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given; these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and GEM distributions with parameters 1/4 and 1/2. 1. Introduction and
Bayesian inference via classes of normalized random measures
 ICER Working Papers  Applied Mathematics Series 52005, ICER  International Centre for Economic Research
, 2005
"... One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by ..."
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Cited by 9 (2 self)
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One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by an appropriate normalization. Specifically we achieve explicit and tractable forms of the posterior and the marginal distributions, including an explicit and easily used description of generalizations of the important BlackwellMacQueen Pólya urn distribution. Such simplifications are achieved by the use of a latent variable which admits quite interesting interpretations which allow to gain a better understanding of the behaviour of these random probability measures. It is noteworthy that these models are generalizations of models considered by Kingman (1975) in a nonBayesian context. Such models are known to play a significant role in a variety of applications including genetics, physics, and work involving random mappings and assemblies. Hence our analysis is of utility in those contexts as well. We also show how our results may be applied to Bayesian mixture models and describe computational schemes which are generalizations of known efficient methods for the case of the Dirichlet process. We illustrate new examples of processes which can play the role of priors for Bayesian nonparametric inference and finally point out some interesting connections with the theory of generalized gamma convolutions initiated by Thorin and further developed by Bondesson. 1
A modified lookdown construction for the XiFlemingViot process with mutation and populations with recurrent bottlenecks
 ALEA
, 2009
"... populations with recurrent bottlenecks ..."
A tale of three couplings: PoissonDirichlet and GEM approximations for random permutations
 Combin. Probab. Comput
, 2006
"... Abstract. For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmi ..."
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Abstract. For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the PoissonDirichlet process can be coupled so that zero is the limit of the expected ℓ1 distance between the process of cycle length proporortions and the PoissonDirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence. One of the couplings we consider has an analog for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the “scale invariant spacing lemma ” for the scale invariant Poisson processes, proved in this paper.
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
Random Partitioning Problems Involving Poisson Point Processes On The Interval
, 2004
"... Suppose some random resource (energy, mass or space) χ ≥ 0 is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume Eχ = θ < ∞ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can natural ..."
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Suppose some random resource (energy, mass or space) χ ≥ 0 is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume Eχ = θ < ∞ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can naturally be identified with the study of infinitely divisible random variables with Lévy measure concentrated on the interval. Special emphasis is put on these special partitioning models in the PoissonKingman class. The masses attached to the atoms of such partitions are sorted in decreasing order. Considering nearestneighbors spacings yields a partition of unity which also deserves special interest. For such partition models, various statistical questions are addressed among which: correlation structure, cumulative energy of the first K largest items, partition function, threshold and covering statistics, weighted partition, Rényi’s, typical and sizebiased fragments size. Several physical images are supplied. When the unbounded Lévy measure of χ is θx −1 ·I (x ∈ (0,1))dx, the spacings partition has GriffithsEngenMcCloskey or GEM(θ) distribution and χ follows Dickman distribution. The induced partition models have many remarkable peculiarities which are outlined. The case with finitely many (Poisson) fragments in the partition law is also briefly addressed. Here, the Lévy measure is bounded.
Corners and Records of the Poisson Process in Quadrant
, 709
"... The scaleinvariant spacings lemma due to Arratia, Barbour and Tavaré establishes the distributional identity of a selfsimilar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the un ..."
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The scaleinvariant spacings lemma due to Arratia, Barbour and Tavaré establishes the distributional identity of a selfsimilar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant.
doi:10.1017/S0963548305007054 Printed in the United Kingdom A Tale of Three Couplings: Poisson–Dirichlet and GEM Approximations for Random Permutations
, 2005
"... For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the secondlongest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For so ..."
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For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the secondlongest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected 1 distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence. One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scaleinvariant spacing lemma ’ for the scaleinvariant Poisson processes, proved in this paper. 1.
Elect. Comm. in Probab. 13 (2008), 187–193 ELECTRONIC COMMUNICATIONS in PROBABILITY CORNERS AND RECORDS OF THE POISSON PRO CESS IN QUADRANT
, 2007
"... The scaleinvariant spacings lemma due to Arratia, Barbour and Tavare ́ establishes the distributional identity of a selfsimilar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the ..."
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The scaleinvariant spacings lemma due to Arratia, Barbour and Tavare ́ establishes the distributional identity of a selfsimilar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant. 1