Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Cannings model to Brownian motion conditioned on local time profile

Xiaodan Li, Chengshi Wang, Yushu Zheng

TL;DR

This work develops a comprehensive scaling-limit theory for genealogies generated by Cannings models, including time-inhomogeneous versions with varying generation sizes. By encoding Cannings trees with contour and height processes and linking their limits to a time-changed Brownian motion conditioned on a local-time profile, the authors connect discrete population genealogies to the continuum object $W^{\ell^{\sigma}}$, a process initially constructed by Warren–Yor and Aldous. The proof combines Aldous’ equivalence for contour convergence with a novel sequential coming-down-from-infinity analysis of a coalescent process, yielding tightness and finite-dimensional convergence that culminate in joint convergence of contour and height to the conditioned Brownian limit and its subtree distributions. The results extend the classical Cannings-Wright–Fisher/Moran frameworks, illustrating how inhomogeneity and local-time conditioning influence genealogies, and they open paths to studying perturbations, immigration, and other population-model generalizations within the same continuum-tree paradigm.

Abstract

We study the scaling limits of genealogical trees arising from Cannings models. Under suitable moment conditions, we show that the rescaled contour and height functions converge to a time change of Brownian motion conditioned on a given local time profile. This conditioned Brownian motion is a self-interacting diffusion constructed independently by Warren--Yor (1998) and Aldous (1998). A key ingredient in our proof is a sequential version of the coming-down-from-infinity property.

From Cannings model to Brownian motion conditioned on local time profile

TL;DR

This work develops a comprehensive scaling-limit theory for genealogies generated by Cannings models, including time-inhomogeneous versions with varying generation sizes. By encoding Cannings trees with contour and height processes and linking their limits to a time-changed Brownian motion conditioned on a local-time profile, the authors connect discrete population genealogies to the continuum object , a process initially constructed by Warren–Yor and Aldous. The proof combines Aldous’ equivalence for contour convergence with a novel sequential coming-down-from-infinity analysis of a coalescent process, yielding tightness and finite-dimensional convergence that culminate in joint convergence of contour and height to the conditioned Brownian limit and its subtree distributions. The results extend the classical Cannings-Wright–Fisher/Moran frameworks, illustrating how inhomogeneity and local-time conditioning influence genealogies, and they open paths to studying perturbations, immigration, and other population-model generalizations within the same continuum-tree paradigm.

Abstract

We study the scaling limits of genealogical trees arising from Cannings models. Under suitable moment conditions, we show that the rescaled contour and height functions converge to a time change of Brownian motion conditioned on a given local time profile. This conditioned Brownian motion is a self-interacting diffusion constructed independently by Warren--Yor (1998) and Aldous (1998). A key ingredient in our proof is a sequential version of the coming-down-from-infinity property.

Paper Structure

This paper contains 36 sections, 50 theorems, 276 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and main results
  3. Related works
  4. Proof outline
  5. Organization of the paper
  6. Preliminaries
  7. Cannings tree
  8. Foundations of tree encoding and convergence
  9. Geometric trees
  10. Encoding compact real trees by bridge functions
  11. Convergence of contour functions
  12. Conditioned Brownian motion and its subtree distribution
  13. Canonical version of $W^\ell$
  14. Distribution of $\mathcal{T}^\ell_k$
  15. Distribution of uniform $k$-point subtrees induced by $W^{\ell^\sigma}_{\alpha^{-1}(t)}$
  16. ...and 21 more sections

Key Result

Theorem 1.1

Assume H1--H2. Define $\alpha(t) :=\frac{1}{4}\int_0^t\sigma(W^{\ell^\sigma}_s)^2\, \mathrm{d} s$. Let $\alpha^{-1}(t)$ be the inverse of $\alpha(t)$We will verify in Lemma lem:alpha that $\alpha$ is a.s. strictly increasing, so $\alpha^{-1}(t)$ is well-defined.. Then in the product space $(D[0, \infty))^2$ equipped with the product Skorokhod topology (where the functions on both sides are extend

Figures (1)

  • Figure 1: An illustration of the coalescent construction of the uniform $6$-point subtree. The vertical axis represents time decreasing from top to bottom. Each leaf $i=1,\ldots, 6$ is placed at height $x_i$ equal to its birth time, and each $b_{i,j}$ denotes the branching point at which the ancestral clusters of $i$ and $j$ merge. The final merge occurs at height $0$, which corresponds to the root.

Theorems & Definitions (109)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: Cannings tree
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Uniform $k$-point subtree induced by $g$
  • Remark 2.5
  • Theorem 2.6
  • ...and 99 more