CRITICAL TREES ARE NEITHER TOO SHORT NOR TOO FAT

Louigi Addario-Berry; Serte Donderwinkel; Igor Kortchemski

doi:10.5802/ahl.231

CRITICAL TREES ARE NEITHER TOO SHORT NOR TOO FAT

Louigi Addario-Berry
, Serte Donderwinkel
, Igor Kortchemski

McGill University
ICS/University of Groningen
ETH Zurich

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise height and width a symptotics when the trees’ offspring distributions lie within the domain of attraction of a Cauchy distribution and satisfy a local regularity condition. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

Translated title of the contribution	LES ARBRES CRITIQUES NE SONT NI TROP COURTS NI TROP GROS
Original language	English
Pages (from-to)	113-149
Number of pages	37
Journal	Annales Henri Lebesgue
Volume	8
DOIs	https://doi.org/10.5802/ahl.231
Publication status	Published - 1 Jan 2025

Keywords

Cauchy distribution
Random trees
Tail bounds

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10.5802/ahl.231

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abstract = "We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaym{\'e} trees. Our bounds are optimal at this level of generality. We also obtain precise height and width a symptotics when the trees{\textquoteright} offspring distributions lie within the domain of attraction of a Cauchy distribution and satisfy a local regularity condition. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaym{\'e} trees.",

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AU - Kortchemski, Igor

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N2 - We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise height and width a symptotics when the trees’ offspring distributions lie within the domain of attraction of a Cauchy distribution and satisfy a local regularity condition. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

AB - We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise height and width a symptotics when the trees’ offspring distributions lie within the domain of attraction of a Cauchy distribution and satisfy a local regularity condition. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

KW - Cauchy distribution

KW - Random trees

KW - Tail bounds

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JO - Annales Henri Lebesgue

JF - Annales Henri Lebesgue

ER -

CRITICAL TREES ARE NEITHER TOO SHORT NOR TOO FAT

Abstract

Keywords

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