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12
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 221 (37 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 17 (2 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Asymptotics of Poisson approximation to random discrete distributions: an analytic approach
 Advances in Applied Probability
, 1998
"... this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the r ..."
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Cited by 13 (9 self)
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this paper, we shall describe the asymptotic behaviors of several distances of Poisson approximation to a wide class of discrete distributions covering many examples from number theory, combinatorics and arithmetic semigroups. Our aim is to show that whenever (analytic) generating functions of the random variables in question are available, complexanalytic methods can be used to derive precise asymptotic results for the five distances above. Actually, we shall consider the following generalized distances: let ff ? 0 be a fixed positive number, (X; Y ) = FM (X; Y ) = (X; Y ) = sup K (X; Y ) = sup M (X; Y ) = jP(X = j) \Gamma P(Y = j) Note that d TV = d M . Besides the case ff = 1 (and ff = 1=2 for d M ), only the case d TV was previously studied by Franken [39] for Poisson approximation to the sum of independent but not identically distributed Bernoulli random variables. We take these quantities as our measures of degree of nearness of Poisson approximation, some of which may be interpreted as certain norms in suitable space as many authors did (cf. [12, 22, 23, 74, 96]). For a large class of discrete distributions, we shall derive an asymptotic main term together with an error estimate for each of these distances. Our results are thus "approximation theorems" rather than "limit theorems". The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]). Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall di...
Random permutations: some grouptheoretic aspects
 257–262. SIZE AND METRIC DIMENSION Page 33 of 34
, 1993
"... The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered ..."
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Cited by 11 (2 self)
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The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions). 1. Permutations lying in a transitive subgroup Sn and An denote the symmetric and alternating groups on the set X = {I,.... n}. A subgroup G of S " is transitive if, for all i, j E X, there exists g E G with ig ~ j. In a preliminary version of this paper, we asked the following question: Question 1.1. Is it true that,/or almost all permutations g E Sn. the only transitive subgroups containing g are Sn and (possihly) An? Here, of course, 'almost all g E S " have property P ' means 'the proportion of elements of S " not having property P tends to 0 as n> ex'. An affirmative answer to this question was given by Luczak and Pyber, in [15]. We will
The PoissonDirichlet Distribution And Its Relatives Revisited
, 2001
"... The PoissonDirichlet distribution and its marginals are studied, in particular the largest component, that is Dickman's distribution. Sizebiased sampling and the GEM distribution are considered. Ewens sampling formula and random permutations, generated by the Chinese restaurant process, are also i ..."
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Cited by 10 (0 self)
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The PoissonDirichlet distribution and its marginals are studied, in particular the largest component, that is Dickman's distribution. Sizebiased sampling and the GEM distribution are considered. Ewens sampling formula and random permutations, generated by the Chinese restaurant process, are also investigated. The used methods are elementary and based on properties of the finitedimensional Dirichlet distribution. Keywords: Chinese restaurant process; Dickman's function; Ewens sampling formula; GEM distribution; Hoppe's urn; random permutations; residual allocation models; sizebiased sampling ams 1991 subject classification: primary 60g57 secondary 60c05, 60k99 Running title: The PoissonDirichlet distribution revisited 1
A note on the approximation of perpetuities
 In Proceedings of 2007 Conference on Analysis of Algorithms, (AofA’07) Juanlespins
, 2007
"... We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier al ..."
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Cited by 6 (4 self)
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We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated.
Poissondirichlet branching random walks. 2010. Available via http://arxiv.org/abs/1012.2544
"... ABSTRACT. We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v1, v2,...) where each vj is the sum of j independent Exponential(1) random variables and the different vi need not be independe ..."
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Cited by 2 (1 self)
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ABSTRACT. We determine, to within O(1), the expected minimal position at level n in certain branching random walks. The walks under consideration have displacement vector (v1, v2,...) where each vj is the sum of j independent Exponential(1) random variables and the different vi need not be independent. In particular, our analysis applies to the PoissonDirichlet branching random walk and to the Poissonweighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within O(1). 1.
A Markov process associated with plotsize distribution in Czech Land Registry and its numbertheoretic properties
 J. Phys. A: Math. Theor
, 2008
"... The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing th ..."
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Cited by 2 (0 self)
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The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and buildup areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case. The distribution of commodities is an important research topic in economy – see [CC07] for an extensive literature overview. In this letter we focus on a particular case, the allocation of land representing a nonconsumable commodity, and a way in which the distribution is reached. Generally speaking, it results from a process of random commodity exchanges between agents in the situation when the aggregate commodity volume is conserved, in other words, one deals with pure trading which leads commodity redistribution. Models of this type were recently intensively discussed [SGG06] and are usually referred to as kinetic exchange models. Our approach here will be different being based on the concept known as
PRIME CHAINS AND PRATT TREES
"... ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains wit ..."
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Cited by 1 (0 self)
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ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x. 1.
Methodol Comput Appl Probab (2008) 10:507–529 DOI 10.1007/s110090079059x Approximating Perpetuities
, 2007
"... Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the e ..."
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Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well.