Light-tail Cannings convergence to tree-valued Fleming–Viot dynamics

For each $N\in\mathbb N$, let $\{V^N_1,\ldots,V^N_N\}$ be the exchangeable offspring variables of a Cannings model, with $\sum_{i=1}^N V_i^N=N$, and let $\mathcal V^N$ be the corresponding tree-valued Cannings dynamics. Assume

$$\lim_{N\to\infty}\frac{\mathbf E[(V_1^N)^2]}{N}=0,$$

and

$$\lim_{N\to\infty}\frac{1}{N}\frac{\mathbf E[(V_1^N)^3]}{\mathbf E[(V_1^N)^2]}=0.$$

Light-tail Cannings conjecture. Under a suitable time rescaling, $(\mathcal V^N)_{N\in\mathbb N}$ converges weakly to the tree-valued Fleming–Viot dynamics in the Skorohod topology on $\mathcal D_{\mathbb U}([0,\infty))$ as $N\to\infty$. This extends the corresponding measure-valued convergence of Cannings genealogies to convergence of genealogical trees.