## The brownian excursion multi-dimensional local time density (1999)

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Venue: | Journal of Applied Probability |

Citations: | 11 - 9 self |

### BibTeX

@ARTICLE{Gittenberger99thebrownian,

author = {Bernhard Gittenberger and Guy Louchard},

title = {The brownian excursion multi-dimensional local time density},

journal = {Journal of Applied Probability},

year = {1999},

pages = {36--350}

}

### Years of Citing Articles

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### Abstract

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.

### Citations

129 |
On the altitude of nodes in random trees
- Meir, Moon
- 1978
(Show Context)
Citation Context ...lations to theoretical computer science Another BE application is the number of nodes at some level in a random tree. Consider a simply generated random tree (according to the notion of Meir and Moon =-=[22]-=-) or, equivalently, the family tree of a Galton-Watson branching process conditioned on the total progeny. Then BE appears as the weak limit of the contour process of this tree, i.e. the process const... |

107 | Processus Stochastiques et Mouvement Brownien - Levy, P - 1948 |

77 |
The continuum random tree. II. An overview
- Aldous
- 1990
(Show Context)
Citation Context ...otal progeny. Then BE appears as the weak limit of the contour process of this tree, i.e. the process constructed of the distances of the nodes from the root when traversing the tree (for details see =-=[1]-=- or [9]). The generation sizes of such branching processes converge weakly to BE local time. Since we will use the correspondence between branching processes and local time later (cf. Section 2.3) we ... |

63 | On the profile of random trees
- Drmota, Gittenberger
- 1997
(Show Context)
Citation Context ...ote that for an excursion of length ` we have:s+ (`; a) d j p ` + (a= p `). See for instance Getoor and Sharpe [8], Knight [16], Cohen and Hooghiemstra [3], Hooghiemstra [11], Drmota and Gittenberger =-=[5]-=-. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can be found in [3] and [5]. In both papers indirec... |

43 |
Kac’s formula, Lévy’s local time and Brownian excursion
- Louchard
- 1984
(Show Context)
Citation Context ...ent to a BE (see Louchard et al [21]). So the distribution of the size of this structure is asymptotically given, on [0; 1], by the BE local time. For other applications of Kac's formula see Louchard =-=[19]-=-. Let us mention that Brownian motion (and more general Gaussian processes) has many applications in data structures and algorithms analysis. See for instance Louchard [20] where other references can ... |

32 | The distribution of the maximum Brownian excursion - Kennedy - 1976 |

31 | Fundamentals of the Average Case Analysis of Particular Algorithms - KEMP - 1984 |

30 |
Brownian excursions, trees and measure-valued branching processes
- GALL
- 1991
(Show Context)
Citation Context ...l time as limit process (see [4]). We also want to mention that BE is used as model of continuum trees, a concept introduced by Aldous [1]. Readers interested in this area should for instance consult =-=[7]-=-. 1.1.3 Dynamical algorithms The priority queue in Knuth's model is combinatorially equivalent to a Markov Stack, which is asymptotically equivalent to a BE (see Louchard et al [21]). So the distribut... |

25 |
Excursions in Brownian motion
- Chung
- 1976
(Show Context)
Citation Context ... L ff (p(t; x; y)) = exp(\Gamma p \Deltajx \Gamma yj) p \Delta (2) where the Laplace transform is taken with respect to t. The maxima of the meander and excursion processes have been studied by Chung =-=[2]-=-, Equations (3.17), (4.9), p. 169 and 177; and Kennedy [15, Theorem 1]. They obtain E 0 max 0ur Z(u)sjL \Gamma (t) = r = f 1 ( 2 =2r) where f 1 (x) = +1 X n=\Gamma1 (\Gamma1) n exp(\Gamman 2 x) and E ... |

22 |
Excursions of Brownian motion and Bessel processes
- Getoor, Sharpe
- 1979
(Show Context)
Citation Context ...andard scaled excursion X at a, denoted bys+ (a), have been studied by several authors (note that for an excursion of length ` we have:s+ (`; a) d j p ` + (a= p `). See for instance Getoor and Sharpe =-=[8]-=-, Knight [16], Cohen and Hooghiemstra [3], Hooghiemstra [11], Drmota and Gittenberger [5]. Several representations of the one dimensional Brownian Excursion local time density are known. Results for t... |

18 |
Marking in combinatorial constructions: Generating functions and limiting distributions
- Drmota, Soria
- 1995
(Show Context)
Citation Context ... u b i+1 (z; u) = z 1 \Gamma b i (z; u) (9) and a(z) = X n0 b n z n = 1 \Gamma p 1 \Gamma 4z 2 and using standard combinatorial techniques (readers not familiar with these techniques may consult e.g. =-=[6]-=-) we can write down the generating function of these numbers in the form g(z; u 1 ; : : : ; u d ) = X k 1 ;:::;k d ;n0 b (r 1 \Delta\Delta\Deltar d ) k 1 \Delta\Delta\Deltak d n u k1 1 \Delta \Delta \... |

15 | The distribution of nodes of given degree in random trees
- Drmota, Gittenberger
- 1999
(Show Context)
Citation Context ...BE local time. Furthermore, it should be mentioned that not only nodes or leaves at some level in a random tree but also nodes of an arbitrarily given degree yield BE local time as limit process (see =-=[4]-=-). We also want to mention that BE is used as model of continuum trees, a concept introduced by Aldous [1]. Readers interested in this area should for instance consult [7]. 1.1.3 Dynamical algorithms ... |

13 |
Brownian excursion, the M/M/1 queue and their occupation times
- Cohen, Hooghiemstra
- 1981
(Show Context)
Citation Context ...ys+ (a), have been studied by several authors (note that for an excursion of length ` we have:s+ (`; a) d j p ` + (a= p `). See for instance Getoor and Sharpe [8], Knight [16], Cohen and Hooghiemstra =-=[3]-=-, Hooghiemstra [11], Drmota and Gittenberger [5]. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can... |

12 |
Ch.: The asymptotic contour process of a binary tree is a Brownian excursion
- Gutjahr, Pflug
- 1992
(Show Context)
Citation Context ...branching processes and local time later (cf. Section 2.3) we do not go into details now. Note that binary trees also belong to the class of simply generated trees and that recently Gutjahr and Pflug =-=[10] showed th-=-at the process constructed of the leaf heights during tree traversal converges to BE. The corresponding "local" result can be found in [5]. Binary trees play an important role in theoretical... |

12 |
Random walks, Gaussian processes and list structures. Theoret
- Louchard
- 1987
(Show Context)
Citation Context ... Kac's formula see Louchard [19]. Let us mention that Brownian motion (and more general Gaussian processes) has many applications in data structures and algorithms analysis. See for instance Louchard =-=[20]-=- where other references can be found. This paper is organized as follows. In Sec. 2 we summarize basic notations and known results. Some preliminary formulas based on Kac's formula are given in Sec. 3... |

10 |
On the explicit form of the density of Brownian excursion local time
- Hooghiemstra
(Show Context)
Citation Context ... studied by several authors (note that for an excursion of length ` we have:s+ (`; a) d j p ` + (a= p `). See for instance Getoor and Sharpe [8], Knight [16], Cohen and Hooghiemstra [3], Hooghiemstra =-=[11]-=-, Drmota and Gittenberger [5]. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can be found in [3] an... |

10 | Diffusion processes and their sample paths,” 2nd corrected printing - Itô, McKean - 1974 |

7 | A course of modern analysis, reprint of the 4th ed - Whittaker, Watson - 1927 |

5 |
On the excursion process of Brownian motion
- KNIGHT
- 1980
(Show Context)
Citation Context ...d excursion X at a, denoted bys+ (a), have been studied by several authors (note that for an excursion of length ` we have:s+ (`; a) d j p ` + (a= p `). See for instance Getoor and Sharpe [8], Knight =-=[16]-=-, Cohen and Hooghiemstra [3], Hooghiemstra [11], Drmota and Gittenberger [5]. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimens... |

5 |
Dynamic algorithms in D.E. Knuth's Model : a Probabilistic Analysis
- LOUCHARD, SCHOTT, et al.
- 1992
(Show Context)
Citation Context ...or instance consult [7]. 1.1.3 Dynamical algorithms The priority queue in Knuth's model is combinatorially equivalent to a Markov Stack, which is asymptotically equivalent to a BE (see Louchard et al =-=[21]-=-). So the distribution of the size of this structure is asymptotically given, on [0; 1], by the BE local time. For other applications of Kac's formula see Louchard [19]. Let us mention that Brownian m... |

4 |
On the pro of random trees
- DRMOTA, GITTENBERGER
- 1997
(Show Context)
Citation Context ... that for an excursion of length ` we have: + (`; a) d p ` + (a= p `). See for instance Getoor and Sharpe [8], Knight [16], Cohen and Hooghiemstra [3], Hooghiemstra [11], Drmota and Gittenberger [5]=-=-=-. Several representations of the one dimensional Brownian Excursion local time density are known. Results for the two dimensional local time density can be found in [3] and [5]. In both papers indirec... |

4 |
On the distribution of the number of vertices in layers of random trees
- Takács
- 1991
(Show Context)
Citation Context ...√ π ·ρ1R] √ dv1 · ·dvr−1 2ρ1 � 1 − e−2 √ �r−1 ·ρ1 2 √ ·ρ1 µr = µr(1) = 1 � e 2πi S α f7(α)dα dα (42) Identification with (42) is immediate. Using the same trick, we can now recover equ (47) of Takács =-=[23]-=-. Indeed this expression gives E[e −β1τ + (ρ1) ] = ∞� 1 + 2 Set w = v − k and But we know that k=1 (2β1ρ1) k (k − 1)! � ∞ k f8(u) := 2 Lα (1 − 4ρ 2 1v 2 )(v − k) k−1 exp[−2ρ 2 1v 2 − 2ρ1(v − k)β1]dv. ... |

2 |
On the average oscillation of a stack
- KEMP
- 1982
(Show Context)
Citation Context ...n theoretical computer science (see e.g. [17]). Consider, e.g., searching data stored in a binary tree by level order traversal (see [14, pp.82]) of the tree. Then the stack size process (analyzed in =-=[13]-=-) can be described by the leaf height process. Thus the number of downcrossings of the stack size process can be described by BE local time. Furthermore, it should be mentioned that not only nodes or ... |

2 | Diusion Processes and their Sample Paths, 2nd edn - IT, K, et al. - 1974 |

1 |
On the Distribution of the Number of Vertices in Layers of Random Trees
- ACS
- 1991
(Show Context)
Citation Context ...ma2 p \Deltaae 1 2 p \Deltaae 1 ! r\Gamma1 Now, set u = 1, sosr =sr (1) = 1 2i Z S e ff f 7 (ff)dff Identification with (42) is immediate. Using the same trick, we can now recover equ (47) of Tak'acs =-=[23]-=-. Indeed this expression gives E[e \Gammafi 1s+ (ae 1 ) ] = 1 + 2 1 X k=1 (2fi 1 ae 1 ) k (k \Gamma 1)! Z 1 k (1 \Gamma 4ae 2 1 v 2 )(v \Gamma k) k\Gamma1 exp[\Gamma2ae 2 1 v 2 \Gamma 2ae 1 (v \Gamma ... |