arXiv subjects

Michel Nassif

Publications and source records attributed to Michel Nassif.

3 recordsLinked to original sources

arXiv2022

Conditioning (sub)critical L{é}vy trees by their maximal degree: Decomposition and local limit

We study the maximal degree of (sub)critical L{é}vy trees which arise as the scaling limits of Bienaym{é}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{é}vy tree conditioned to have a fixed maximal degree. In the case where the L{é}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{é}vy tree can be obtained by grafting a L{é}vy forest on an independent size-biased L{é}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{é}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.

math.PR↗

arXiv2021

Limiting Behavior Of Additive Functionals On The Stable Tree

We study the shape of the normalized stable Lévy tree $\mathcal{T}$ near its root. We show that, when zooming in at the root at the proper speed with a scaling depending on the index of stability, we get the unnormalized Kesten tree. In particular the limit is described by a tree-valued Poisson point process which does not depend on the initial normalization. We apply this to study the asymptotic behavior of additive functionals of the form \[\mathbf{Z}_{α,β}=\int_{\mathcal{T}} μ(\mathrm{d} x) \int_0^{H(x)} σ_{r,x}^α\mathfrak{h}_{r,x}^β\,\mathrm{d} r\]as $\max(α,β) \to \infty$, where $μ$ is the mass measure on $\mathcal{T}$, $H(x)$ is the height of $x$ and $σ_{r,x}$ (resp. $\mathfrak{h}_{r,x}$) is the mass (resp. height) of the subtree of $\mathcal{T}$ above level $r$ containing $x$. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaym{é}-Galton-Watson trees.

math.PR↗

arXiv2020

Global Regime for General Additive Functionals of Conditioned Bienaym{é}-Galton-Watson Trees

We give an invariance principle for very general additive functionals of conditioned Bienaym{é}-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable L{é}vy tree. This includes the case when the offspring distribution has finite variance (the L{é}vy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.

math.PR↗