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

About 2 years old · traced to 2024

For the stationary distribution of the two-island diffusion, let $(X,Y)$ have distribution

(X,Y) \sim \operatorname{TI}(a_1,a_2,\gamma b_1,\gamma b_2,c,\gamma c,2,2\gamma).

For the stationary distribution of the seed-bank diffusion, let

(X,Y) \sim \operatorname{TI}(a_1,a_2,0,0,c,\gamma c,2,0).

Beta-limit conjecture. As $c\to\infty$ , in the two-island case both $X$ and $Y$ converge in distribution to the beta distribution with parameters $(a_1+b_1,a_2+b_2)$ ; in the seed-bank case both $X$ and $Y$ converge in distribution to the beta distribution with parameters $(a_1,a_2)$ .

The conjecture is motivated by the fact that the two islands become asymptotically indistinguishable and that the first and second moments calculated in the preceding lemma match the corresponding beta moments. The supplied text gives no resolution, so the conjecture remains open.

References

Primary source

Han L. Gan and Maite Wilke-Berenguer, “Stationary distribution approximations of Two-island Wright-Fisher and seed-bank models using Stein's method”, arXiv:2405.18763 (2024).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The published conjecture is unproved, while a reader-written calculation claims the two-island case is true and gives a counterexample to the seed-bank case.

Gan and Wilke-Berenguer (2024) conjectured that, as $c\to\infty$ , both coordinates approach the same beta law: $\operatorname{Beta}(a_1+b_1,a_2+b_2)$ for two islands and $\operatorname{Beta}(a_1,a_2)$ for the seed bank. This is Conjecture $2.7$ .

Known results

The stationary coordinates coalesce: $X-Y\to0$ in probability as $c\to\infty$ .
The calculated first and second moments agree with the proposed beta limits.
The exact stationary distribution is generally unknown.

Posted attempt

A reader-written attempt claims a general limiting theorem: the two-island clause is proved, whereas the seed-bank clause is false and should instead have beta parameters $((1+\gamma)a_1/\gamma,(1+\gamma)a_2/\gamma)$ . It supplies a second-moment counterexample at $a_1=a_2=\gamma=1$ . This is a partial claim about the combined conjecture and has not been independently verified.

Current status (as of August 2026): The published source leaves both clauses open, while an unverified discussion claims the two-island clause and disputes the seed-bank clause; no corroborated resolution is recorded.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

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 .