On weighted depths in random binary search trees

Following the model introduced by Aguech, Lasmar and Mahmoud [Probab. Engrg. Inform. Sci. 21 (2007) 133-141], 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 analyze 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

The size of random fragmentation intervals

Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1-e^{-x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian

The size of random fragmentation intervals

Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability p witch can not depend on the fragment size. The number of stable fragments with sizes less than a given t ≥ 0, denoted by K(t), is introduced and studied. In the second one the probability to split a fragment of size x is p(x) = 1 − e −x. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It’s shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian

On the protected nodes in exponential recursive trees

The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees

ALGORITHMES STOCHASTIQUES (ETUDE ASYMPTOTIQUE DE L'ERREUR)

PARIS-BIUSJ-Mathématiques rech (751052111) / SudocCentre Technique Livre Ens. Sup. (774682301) / SudocSudocFranceF

New characterizations of completely monotone functions and Bernstein functions, a converse to Hausdorff’s moment characterization theorem

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on N0. We give a complete answer to the following question: Can we affirm that a function f is completely monotone (resp. a Bernstein function) if we know that the sequence f(k)kis completely monotone (resp. alternating)? This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences. Keywords: Completely monotone functions, Completely monotone sequences, Bernstein functions, Completely alternating functions, Completely alternating sequences, Hausdorff moment problem, Hausdorff moment sequences, Self-decomposability, 2010 Mathematics Subject Classification: 30E05, 44A10, 44A60, 47A57, 60E05, 60E07, 60B1