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Enumerations Of Trees And Forests Related To Branching Processes And Random Walks
 Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci
, 1997
"... In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p ..."
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Cited by 39 (15 self)
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In a GaltonWatson branching process with offspring distribution (p 0 ; p 1 ; : : :) started with k individuals, the distribution of the total progeny is identical to the distribution of the first passage time to \Gammak for a random walk started at 0 which takes steps of size j with probability p j+1 for j \Gamma1. The formula for this distribution is a probabilistic expression of the Lagrange inversion formula for the coefficients in the power series expansion of f(z) k in terms of those of g(z) for f(z) defined implicitly by f(z) = zg(f(z)). The Lagrange inversion formula is the analytic counterpart of various enumerations of trees and forests which generalize Cayley's formula kn n\Gammak\Gamma1 for the number of rooted forests labeled by a set of size n whose set of roots is a particular subset of size k. These known results are derived by elementary combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies an...
SIMPLY GENERATED TREES, CONDITIONED GALTON–WATSON TREES, RANDOM ALLOCATIONS AND CONDENSATION (EXTENDED ABSTRACT)
, 2012
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Susceptibility in subcritical random graphs
 125207. OF RANDOM GRAPHS WITH GIVEN VERTEX DEGREES 25
"... Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its ..."
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Cited by 5 (4 self)
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Abstract. We study the evolution of the susceptibility in the subcritical random graph G(n, p) as n tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. 1.
TAKÁCS ’ ASYMPTOTIC THEOREM AND ITS APPLICATIONS: A SURVEY
, 712
"... Abstract. The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with ..."
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Cited by 5 (5 self)
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Abstract. The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with
Individual displacements for linear probing hashing with different insertion policies
 ACM Transactions on Algorithms
, 2005
"... Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occ ..."
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Cited by 5 (1 self)
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Abstract. We study the distribution of the individual displacements in hashing with linear probing for three different versions: First Come, Last Come and Robin Hood. Asymptotic distributions and their moments are found when the the size of the hash table tends to infinity with the proportion of occupied cells converging to some α, 0 < α < 1. (In the case of Last Come, the results are more complicated and less complete than in the other cases.) We also show, using the diagonal Poisson transform studied by Poblete, Viola and Munro, that exact expressions for finite m and n can be obtained from the limits as m, n → ∞. We end with some results, conjectures and questions about the shape of the limit distributions. These have some relevance for computer applications. 1.
SUSCEPTIBILITY IN INHOMOGENEOUS RANDOM GRAPHS
"... Abstract. We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives ..."
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Cited by 2 (2 self)
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Abstract. We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples. 1.
An elementary proof of the hitting time theorem
 American Mathematical Monthly
, 2008
"... In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a leftcontinuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at tim ..."
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In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a leftcontinuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at time n, is equal to k/n. Here, a walk is called leftcontinuous when its steps are bounded from below by −1. We start by introducing some notation. Let Pk denote the law of a random walk starting at k ≥ 0, let {Yi}∞i=1 be the independent and identically distributed (i.i.d.) steps of the random walk, let Sn = k + Y1 + · · · + Yn be the position of the random walk starting at k after n steps, and let T0 = inf{n: Sn = 0} (1) denote the walk’s first hitting time of the origin. Clearly, T0 = 0 when the walker starts at the origin. Then, the hitting time theorem is the following result: Theorem 1 (Hitting time theorem). For a random walk starting at k ≥ 1 with i.i.d. steps {Yi}∞i=1 satisfying Yi ≥ −1 almost surely, the distribution of T0 under Pk is given by
LIMIT DISTRIBUTIONS FOR QUEUES AND RANDOM ROOTED TREES
"... In this paper several limit theorems are proved for the fluctuations of the queue size during the initial busy period of a queuing process with one server. These theorems are used to find the solutions of various problems connected with the heights and widths of random rooted trees. Key words: heigh ..."
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In this paper several limit theorems are proved for the fluctuations of the queue size during the initial busy period of a queuing process with one server. These theorems are used to find the solutions of various problems connected with the heights and widths of random rooted trees. Key words: heights and widths. Singleserver queues, queue size, random trees,