Beta-limit conjecture for stationary two-island and seed-bank diffusions

The two-island assertion is true, but the seed-bank assertion is false. A corrected general limit theorem identifies both cases.

Consider the source's stationary diffusion with generator

$$\begin{aligned} \mathcal L_cf(x,y) ={}&[a_1-(a_1+a_2)x+c(y-x)]f_x +\frac{\alpha}{2}x(1-x)f_{xx}\\ &+[b_1-(b_1+b_2)y+\gamma c(x-y)]f_y +\frac{\beta}{2}y(1-y)f_{yy}. \end{aligned}$$

Let $\gamma>0$, $u_i=\gamma a_i+b_i$, and $K=\gamma^2\alpha+\beta$, with $u_1,u_2,K>0$. Then as $c\to\infty$,

$$\boxed{ (X_c,Y_c)\Longrightarrow(Z,Z), \qquad Z\sim\operatorname{Beta}\left( \frac{2(1+\gamma)(\gamma a_1+b_1)}{\gamma^2\alpha+\beta}, \frac{2(1+\gamma)(\gamma a_2+b_2)}{\gamma^2\alpha+\beta} \right). } \tag{1}$$

To prove this, stationarity applied to $(x-y)^2$ gives

$$\mathbb E(X_c-Y_c)^2=O(c^{-1}).$$

Now set

$$W_c=\frac{\gamma X_c+Y_c}{1+\gamma}.$$

Migration cancels exactly in $\mathcal L_ch(W_c)$. Therefore every subsequential weak limit $W$ satisfies, for every $h\in C^2[0,1]$,

$$\mathbb E\left[ \{u_1-(u_1+u_2)W\}h'(W) +\frac{K}{2(1+\gamma)} W(1-W)h''(W) \right]=0. \tag{2}$$

Taking $h(w)=w^r$ determines all moments:

$$m_r =\frac{\theta_1+r-1}{\theta_1+\theta_2+r-1}m_{r-1}, \qquad \theta_i=\frac{2(1+\gamma)u_i}{K}.$$

Compactness of $[0,1]$ makes these moments determining, so (2) gives the beta law in (1), and diagonal collapse upgrades this to joint convergence.

For the conjectured two-island specialization, the generator parameters are

$$(b_1^{\rm gen},b_2^{\rm gen},\alpha,\beta) =(\gamma b_1,\gamma b_2,2,2\gamma).$$

Formula (1) correctly gives

$$Z\sim\operatorname{Beta}(a_1+b_1,a_2+b_2),$$

proving the first clause.

For the seed-bank specialization, however,

$$(b_1,b_2,\alpha,\beta)=(0,0,2,0),$$

and (1) gives the actual limit

$$\boxed{ Z\sim \operatorname{Beta}\left( \frac{1+\gamma}{\gamma}a_1, \frac{1+\gamma}{\gamma}a_2 \right), }$$

not the asserted $\operatorname{Beta}(a_1,a_2)$.

A completely explicit counterexample follows by taking $a_1=a_2=\gamma=1$. Stationarity for $x,y,x^2,xy,y^2$ yields, exactly for every $c>0$,

$$\mathbb EX_c=\mathbb EY_c=\frac12,$$ $$\mathbb EX_c^2=\frac{3c+4}{10c+12}, \qquad \mathbb E[X_cY_c]=\mathbb EY_c^2 =\frac{3(c+1)}{10c+12}.$$

Hence

$$\lim_{c\to\infty}\mathbb EX_c^2=\frac3{10},$$

whereas the claimed $\operatorname{Beta}(1,1)$ limit has second moment $1/3$. Since $x^2$ is bounded and continuous, the claimed convergence is impossible. The correct limit is $\operatorname{Beta}(2,2)$.

Thus the original combined conjecture is false; its first clause holds, and its second clause requires the missing factor $(1+\gamma)/\gamma$ in both beta parameters.

Source: Gan and Wilke-Berenguer, Stationary distribution approximations of Two-island Wright-Fisher and seed-bank diffusions using Stein's method, equation (1.1), Lemma 2.6, and Conjecture 2.7, https://arxiv.org/abs/2405.18763 .