scholarly journals On the explicit form of the density of Brownian excursion local time

1982 ◽  
Vol 84 (1) ◽  
pp. 127-127 ◽  
Author(s):  
G. Hooghiemstra
2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.


1999 ◽  
Vol 36 (2) ◽  
pp. 350-373 ◽  
Author(s):  
Bernhard Gittenberger ◽  
Guy Louchard

Expressions for the multi-dimensional densities of Brownian excursion 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 Galton–Watson trees.


Author(s):  
Gabriel Berzunza Ojeda ◽  
Svante Janson

Abstract It is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.


1984 ◽  
Vol 21 (3) ◽  
pp. 479-499 ◽  
Author(s):  
G. Louchard

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.


1984 ◽  
Vol 21 (03) ◽  
pp. 479-499 ◽  
Author(s):  
G. Louchard

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.


2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger

International audience It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ ^2 are asymptotically given by (σ √n)^pm_p where m_p 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.


1999 ◽  
Vol 36 (02) ◽  
pp. 350-373 ◽  
Author(s):  
Bernhard Gittenberger ◽  
Guy Louchard

Expressions for the multi-dimensional densities of Brownian excursion 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 Galton–Watson trees.


2015 ◽  
Vol 37 (4) ◽  
pp. 303-315 ◽  
Author(s):  
Pham Chi Vinh ◽  
Nguyen Thi Khanh Linh ◽  
Vu Thi Ngoc Anh

This paper presents  a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.


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