Results 1  10
of
11
On The Contour Of Random Trees
 SIAM J. Discrete Math
"... Two stochastic processes describing the contour of simply generated random trees are studied: the contour process as defined by Gutjahr and Pflug [9] and the traverse process constructed of the node heights during preorder traversal of the tree. Using multivariate generating functions and singulari ..."
Abstract

Cited by 63 (20 self)
 Add to MetaCart
Two stochastic processes describing the contour of simply generated random trees are studied: the contour process as defined by Gutjahr and Pflug [9] and the traverse process constructed of the node heights during preorder traversal of the tree. Using multivariate generating functions and singularity analysis the weak convergence of the contour process to Brownian excursion is shown and a new proof of the analogous result for the traverse process is obtained. 1.
Analytic Variations On The Airy Distribution
, 2001
"... . The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curio ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
. The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders 1; 3; 5; &c, as well as + 1 3 ; 5 3 ; 11 3 ; &c. and 7 3 ; 13 3 ; 19 3 ; &c . Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on \nonprobabilistic" arguments like analytic continuation. A byproduct of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +1, and power symmetric functions of the zeros k of Ai(z). For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The ...
Invariance principles for nonuniform random mappings and trees
 ASYMPTOTIC COMBINATORICS WITH APPLICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases ..."
Abstract

Cited by 11 (9 self)
 Add to MetaCart
In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases, in this paper we formulate a sampling invariance principle in terms of iterates of a fixed number of random elements. We show that the sampling invariance principle implies many, but not all, of the distributional limits implied by the functional invariance principle. We give direct verifications of the sampling invariance principle in two successive generalizations of the uniform case, to pmappings (where elements are mapped to i.i.d. nonuniform elements) and Pmappings (where elements are mapped according to a Markov matrix). We compare with parallel results in the simpler setting of random trees.
The Width of GaltonWatson Trees Conditioned by the Size
, 2004
"... It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a ti ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1.
Strata of random mappings – a combinatorial approach
 Stoch. Proc. Appl
, 1999
"... Abstract. Consider the functional graph of a random mapping from an n–element set into itself. Then the number of nodes in the strata of this graph can be viewed as stochastic process. Using a generating function approach it is shown that a suitable normalization of this process converges weakly to ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Abstract. Consider the functional graph of a random mapping from an n–element set into itself. Then the number of nodes in the strata of this graph can be viewed as stochastic process. Using a generating function approach it is shown that a suitable normalization of this process converges weakly to local time of reflecting Brownian bridge. 1.
THE SHAPE OF UNLABELED ROOTED RANDOM TREES
"... Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the h ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned GaltonWatson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned GaltonWatson process. 1.
On the local time density of the reflecting Brownian bridge
 MR MR1768499 (2001h:60134
, 2000
"... Expressions for the multidimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Expressions for the multidimensional densities of Brownian bridge local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for strata of random mappings.
Noncrossing trees are almost conditioned Galton– Watson trees
 Random Struct. Alg
"... ABSTRACT: A noncrossing tree (NCtree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this paper, we show that NCtrees with size n are conditioned Galton–Watson trees. As corollaries, we ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
ABSTRACT: A noncrossing tree (NCtree) is a tree drawn on the plane having as vertices a set of points on the boundary of a circle, and whose edges are straight line segments that do not cross. In this paper, we show that NCtrees with size n are conditioned Galton–Watson trees. As corollaries, we give the limit law of depthfirst traversal processes and the limit profile of NCtrees. 1
Generalized covariances of multidimensional Brownian Excursion local times
, 2003
"... Expressions for the generalized covariances of multidimensional Brownian excursion local times are derived from corresponding densities transforms. Typical applications are moments of the cost of structures such as M/G/1 queue, Random trees, Markov stack or priority queue in Knuth's model. Brownian ..."
Abstract
 Add to MetaCart
Expressions for the generalized covariances of multidimensional Brownian excursion local times are derived from corresponding densities transforms. Typical applications are moments of the cost of structures such as M/G/1 queue, Random trees, Markov stack or priority queue in Knuth's model. Brownian excursion area and a result of Biane and Yor are also revisited.
THE DEGREE PROFILE OF PÓLYA TREES
"... Abstract. We investigate the profile of random Pólya trees of size n when only nodes of degree d are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreove ..."
Abstract
 Add to MetaCart
Abstract. We investigate the profile of random Pólya trees of size n when only nodes of degree d are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree d1 and d2 in the levels of the tree. 1.