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38
On the Distribution for the Duration of a Randomized Leader Election Algorithm
 Ann. Appl. Probab
, 1996
"... We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressio ..."
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Cited by 38 (10 self)
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We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic fluctuations. In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization, and dePoissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis metho...
Probabilistic bounds on the coefficients of polynomials with only real zeros
, 1997
"... The work of Harper and subsequent authors has shown that finite sequences (a0,..., an) arising from combinatorial problems are often such that the polynomial A(z): = n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first a ..."
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Cited by 32 (0 self)
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The work of Harper and subsequent authors has shown that finite sequences (a0,..., an) arising from combinatorial problems are often such that the polynomial A(z): = n k=0 akz k has only real zeros. Basic examples include rows from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming the ak are nonnegative, A(1)>0 and that A(z) is not constant, it is known that A(z) has only real zeros iff the normalized sequence (a0 A(1),..., an A(1)) is the probability distribution of the number of successes in n independent trials for some sequence of success probabilities. Such sequences (a0,..., an) are also known to be characterized by total positivity of the infinite matrix (ai & j) indexed by nonnegative integers i and j. This papers reviews inequalities and approximations for such sequences, called Polya frequency sequences which follow from their probabilistic representation. In combinatorial examples these inequalities yield a number of improvements of known estimates.
Some Probabilistic Aspects Of Set Partitions
 American Mathematical Monthly
, 1996
"... this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the ..."
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Cited by 27 (1 self)
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this paper, section (1.2) offers an elementary combinatorial proof of Dobinski's formula which seems simpler than other proofs in the literature (Rota [35], Berge [5], p. 44, Comtet [9], p. 211). This argument involves identities whose probabilistic interpretations are brought out later in the paper. 1.1 Notation
Limit theorems for combinatorial structures via discrete process approximations
 RANDOM STRUCTURES AND ALGORITHMS
, 1992
"... Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide e ..."
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Cited by 21 (2 self)
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Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the ErdosTurin Law for the logorder of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of
Clustering in coagulationfragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and ZeroOne law
"... The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=compon ..."
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Cited by 19 (11 self)
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The equilibrium distribution of a reversible coagulationfragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=components) in the case a(k) = kp−1, k ≥ 1, p> 0, where a(k), k ≥ 1 is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic random combinatorial structures (RCS’s) and for RCS’s, corresponding to the case p < 0. 1 Summary. Our main result is a central limit theorem (Theorem 2.4) for the number of groups at steady state for a class of reversible CFP’s and for the corresponding class of RCS’s. In Section 2, we provide a definition of a reversible kCFP admitting interactions of up to k groups, as a generalization of the standard 2CFP. The steady state of the processes considered is fully defined by a parameter function a ≥ 0 on the set of integers. It was observed by Kelly ([11], p. 183) that for all 2 ≤ k ≤ N the kCFP’s have the same invariant measure on the set of partitions of a given
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 16 (8 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...
Large Deviations for Integer Partitions
, 1998
"... We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure ..."
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Cited by 13 (1 self)
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We consider deviations from limit shape induced by uniformly distributed partitions (and strict partitions) of an integer n on the associated Young diagrams. We prove a full large deviation principle, of speed p n. The proof, based on projective limits, uses the representation of the uniform measure on partitions by means of suitably conditioned independent variables.
The PoissonDirichlet distribution and the scaleinvariant Poisson process
 COMBIN. PROBAB. COMPUT
, 1999
"... We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that ..."
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Cited by 12 (4 self)
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We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T, and conditioning on the event that T � 1. Restricting both processes to (0,β] for 0 <β � 1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.