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18
Local limit theorems for finite and infinite urn models
 ANN. PROBAB
, 2008
"... Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. ..."
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Cited by 23 (2 self)
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Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation.
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Sparse graphs: metrics and random models
"... Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the d ..."
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Cited by 13 (1 self)
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Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a sense completely
Renewals for exponentially increasing lifetimes, with an application to digital search
, 2007
"... We show that the number of renewals up to time t exhibits distributional fluctuations as t → ∞ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a probabilistic explanation for the asymptotics of insertion depth in random trees generated by a bitc ..."
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Cited by 9 (3 self)
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We show that the number of renewals up to time t exhibits distributional fluctuations as t → ∞ if the underlying lifetimes increase at an exponential rate in a distributional sense. This provides a probabilistic explanation for the asymptotics of insertion depth in random trees generated by a bitcomparison strategy from uniform input; we also obtain a representation for the resulting family of limit laws along subsequences. Our approach can also be used to obtain rates of convergence.
Asymptotic variance of random symmetric digital search trees
, 2009
"... Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more caref ..."
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Cited by 8 (5 self)
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Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic dePoissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n(log n) 2variance for certain notions of total pathlength is also clarified.
The size of random fragmentation trees
 In preparation
"... Abstract. We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given ..."
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Cited by 6 (0 self)
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Abstract. We consider the random fragmentation process introduced by Kolmogorov, where a particle having some mass is broken into pieces and the mass is distributed among the pieces at random in such a way that the proportions of the mass shared among different daughters are specified by some given probability distribution (the dislocation law); this is repeated recursively for all pieces. More precisely, we consider a version where the fragmentation stops when the mass of a fragment is below some given threshold, and we study the associated random tree. Dean and Majumdar found a phase transition for this process: the number of fragmentations is asymptotically normal for some dislocation laws but not for others, depending on the position of roots of a certain characteristic equation. This parallels the behaviour of discrete analogues with various random trees that have been studied in computer science. We give rigorous proofs of this phase transition, and add further details. The proof uses the contraction method. We extend some previous results for recursive sequences of random variables to families of random variables with a
Convergence of some leader election algorithms
, 2008
"... We start with a set of n players. With some probability P(n, k), we kill n −k players; the other ones stay alive, and we repeat with them. What is the distribution of the number Xn of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some ..."
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Cited by 4 (2 self)
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We start with a set of n players. With some probability P(n, k), we kill n −k players; the other ones stay alive, and we repeat with them. What is the distribution of the number Xn of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions P(n, k), including stochastic monotonicity and the assumption that roughly a fixed proportion α of the players survive in each round. We prove a kind of convergence in distribution for Xn − log 1/α n; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable Z such that d(Xn, ⌈Z+log 1/α n⌉) → 0, where d is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin [7]. We study the latter algorithm further, including numerical
Renewal theory in analysis of tries and strings
, 2009
"... We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes. ..."
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Cited by 4 (1 self)
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We give a survey of a number of simple applications of renewal theory to problems on random strings and tries: insertion depth, size, insertion mode and imbalance of tries; variations for btries and Patricia tries; Khodak and Tunstall codes.
Distributional asymptotics in the analysis of algorithms: Periodicities and discretization
 Discrete Math. Theor. Comput. Sci. Proc. AH
, 2007
"... It is well known that many distributions that arise in the analysis of algorithms have an asymptotically fluctuating behaviour in the sense that we do not have ‘full ’ convergence, but only convergence along suitable subsequences as the size of the input to the algorithm tends to infinity. We are in ..."
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Cited by 3 (3 self)
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It is well known that many distributions that arise in the analysis of algorithms have an asymptotically fluctuating behaviour in the sense that we do not have ‘full ’ convergence, but only convergence along suitable subsequences as the size of the input to the algorithm tends to infinity. We are interested in constructions that display such behaviour via an ordinarily convergent background process in the sense that the periodicities arise from this process by deterministic transformations, typically involving discretization as a decisive step. This leads to structural representations of the resulting family of limit distributions along subsequences, which in turn may give access to their properties, such as the tail behaviour (unsuccessful search in digital search trees) or the dependence on parameters of the algorithm (success probability in a selection algorithm).