Related papers: Exact Hausdorff and packing measures for random se…
We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…
We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.
In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random…
This paper is a detailled study of the coding of real trees by real valued functions that is motivated by probabilistic problems related to continuum random trees. Indeed it is known since the works of Aldous (1993) and Le Gall (1991) that…
We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems…
We study fine properties of the so-called stable trees, which are the scaling limits of critical Galton-Watson trees conditioned to be large. In particular we derive the exact Hausdorff measure function for Aldous' continuum random tree and…
In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing measures of a totally disconnected self-similar set. We evaluate these rates…
We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous…
We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system one-by-one at random, according to a rule which depends on the degree distribution…
In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical…
We study the exact Hausdorff and packing dimensions of the $prime$ $Cantor$ $set$, $\Lambda_P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set…
Decision tree learning is increasingly being used for pointwise inference. Important applications include causal heterogenous treatment effects and dynamic policy decisions, as well as conditional quantile regression and design of…
In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of…
Let $X=\{X(t),t\in\mathbb{R}^N\}$ be a random field with values in $\mathbb{R}^d$. For any finite Borel measure $\mu$ and analytic set $E\subset\mathbb{R}^N$, the Hausdorff and packing dimensions of the image measure $\mu_X$ and image set…
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general…
We establish variational principles for the Hausdorff and packing dimensions of a class of statistically self-affine sponges, including in particular fractal percolation sets obtained from Bara\'nski and Gatzouras-Lalley carpets and…
We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a…
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum…
We study fine properties of L\'evy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. L\'evy trees are the scaling limits of Galton-Watson trees and…
We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic…