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Large deviation principle for the absorption time of the Beta-coalescent via integral functionals

Grégoire Véchambre

TL;DR

<3-5 sentence high-level summary>

Abstract

We study some aspects of the absorption time of the Beta$(a,b)$-Coalescent starting with $n$ blocks. More precisely, when $a>1$, the absorption time is known to converge to infinity as $n$ goes to infinity, and we prove that it satisfies a large deviation principle. When $a \in (0,1)$, it is known that the coalescent comes down from infinity, and we derive bounds for the convergence in the Kolmogorov distance of the distribution of the absorption time as $n$ goes to infinity. To prove our results we introduce a method, inspired from statistical mechanics, that allows to infer the asymptotic behavior of the Laplace transforms of some integral functionals of the Beta-coalescent as the initial number of blocks $n$ goes to infinity. As a by-product of our proofs we also obtain estimates for the record probabilities of the Beta-coalescent.

Large deviation principle for the absorption time of the Beta-coalescent via integral functionals

TL;DR

<3-5 sentence high-level summary>

Abstract

We study some aspects of the absorption time of the Beta-Coalescent starting with blocks. More precisely, when , the absorption time is known to converge to infinity as goes to infinity, and we prove that it satisfies a large deviation principle. When , it is known that the coalescent comes down from infinity, and we derive bounds for the convergence in the Kolmogorov distance of the distribution of the absorption time as goes to infinity. To prove our results we introduce a method, inspired from statistical mechanics, that allows to infer the asymptotic behavior of the Laplace transforms of some integral functionals of the Beta-coalescent as the initial number of blocks goes to infinity. As a by-product of our proofs we also obtain estimates for the record probabilities of the Beta-coalescent.

Paper Structure

This paper contains 17 sections, 23 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Large deviation principle for the absorption time
  4. Integral functionals of the Beta-coalescent
  5. Convergence rate for the absorption time
  6. Organization of the paper
  7. Integral functionals of the Beta-coalescent
  8. Basic properties
  9. Construction of a particular case
  10. Proof of Theorems \ref{['asymptfunctional']} and \ref{['asymptfunctionalbis']}
  11. Large deviation principle for the absorption time
  12. Convergence rate for the absorption time
  13. Connection with an integral functional: Proof of Proposition \ref{['coaltimentonplus1']}
  14. Estimate for the Kolmogorov distance: Proof of Theorem \ref{['boundkolmdistmain']}
  15. The coefficients $\lambda_n(\beta_{a,b})$
  16. ...and 2 more sections

Key Result

Theorem 2.1

For any $a>1, b>0$, the family $(\tau_n/\log n)_{n \geq 1}$ under $\mathbb{P}_{\beta_{a,b}}$ satisfies a large deviation principle with good rate function $\mathcal{I}^2_{a,b}(\cdot)$ and speed $\log n$. More precisely, for any closed set $F \subset \mathbb{R}$, and for any open set $G \subset \mathbb{R}$,

Theorems & Definitions (47)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7: An idea from statistical mechanics
  • Theorem 2.8
  • Proposition 2.9
  • Corollary 2.10
  • ...and 37 more