On Weighted Path Lengths and Distances in Increasing Trees

Kuba, Markus; Panholzer, Alois

doi:10.1017/s0269964807000058

Cited by 3 publications

(7 citation statements)

References 17 publications

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“…The proof of the converse direction establishing (19) is easier. It runs along the same lines upon using the trivial bounds |Ξ k − Ξ k−1 | ≤ 1 and N k ≥ 0.…”

Section: The Weighted Silhouettementioning

confidence: 99%

“…The asymptotic behavior of weighted depths of small nodes is to be compared with the corresponding results in [19]. Here, another phase transition occurs when k = o(n/ √ log n).…”

Section: 1mentioning

confidence: 99%

“…In [1], results about weighted depths of extremal paths have been obtained. Kuba and Panholzer [19], [20] studied the problem in random increasing trees covering the random recursive tree and the random plane-oriented recursive tree. Weighted depths of nodes and the weighted height were also studied by Broutin and Devroye [3] in a more general tree model, which relies on assigning weights to the edges of the tree.…”

Section: Introductionmentioning

confidence: 99%

“…Further, the weighted path length in this model was investigated by Rüschendorf and Schopp [29]. Note that we deviate from the notation introduced in [1] and [19] using the term weighted depth for what is called weighted path length there since we also study a weighted version of the (total) path length of binary search trees.…”

Section: Introductionmentioning

confidence: 99%

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On Weighted Depths in Random Binary Search Trees

2017

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Following the model introduced by Aguech et al. (Probab Eng Inf Sci 21:133-141, 2007), the weighted depth of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. We analyse weighted depths of nodes with given labels, the last inserted node, nodes ordered as visited by the depth first search process, the weighted path length and the weighted Wiener index in a random binary search tree. We establish three regimes of nodes depending on whether the second-order behaviour of their weighted depths follows from fluctuations of the keys on the path, the depth of the nodes or both. Finally, we investigate a random distribution function on the unit interval arising as scaling limit for weighted depths of nodes with at most one child.

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“…The proof of the converse direction establishing (19) is easier. It runs along the same lines upon using the trivial bounds |Ξ k − Ξ k−1 | ≤ 1 and N k ≥ 0.…”

Section: The Weighted Silhouettementioning

confidence: 99%

“…The asymptotic behavior of weighted depths of small nodes is to be compared with the corresponding results in [19]. Here, another phase transition occurs when k = o(n/ √ log n).…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Weighted Depths in Random Binary Search Trees

2017

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“…Two simple examples of sequences Z n of RVs that converge in distribution to Z ∼ GD(θ ) are: [1,Theorem 4.6]. Some recent results where the GD distribution arises as the limit in distribution of certain sequences of RVs are Theorem 1 of [13] and the theorems in [11]. Our example of the class X joins the growing list of instances where the GD distribution is encountered as a limit law (here, in an infinite-dimensional sense).…”

mentioning

confidence: 99%

On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

Covo

2009

J. Appl. Probab.

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let X ε be the sum of jumps not exceeding ε, and let µ(ε) = E[X ε (1)]. We study the question of weak convergence of X ε /µ(ε) as ε ↓ 0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of X ε /ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

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On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

Covo

2009

Journal of Applied Probability

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Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xε be the sum of jumps not exceeding ε, and let µ(ε)=E[Xε(1)]. We study the question of weak convergence of Xε/µ(ε) as ε ↓0, in terms of the limit behavior of µ(ε)/ε. The most interesting case reduces to the weak convergence of Xε/ε to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001).

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On Weighted Path Lengths and Distances in Increasing Trees

Cited by 3 publications

References 17 publications

On Weighted Depths in Random Binary Search Trees

On Weighted Depths in Random Binary Search Trees

On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

On Approximations of Small Jumps of Subordinators with Particular Emphasis on a Dickman-Type Limit

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