U(w;(1/2,1/2))$ as $n\to\infty$ is $w^\intercal \mu\mu^\intercal w - w^\intercal \mu\mu^\intercal {\mathbf{1}}_2 + \mu^\intercal \mu/2 + 1$. Since $\bar y_k\bar y_{k'} = \mu_k\mu_{k'} + O_p(1/\sqrt{n})$ and $\bar y_{k}\bar y = \mu_k(\mu_1 + \mu_2)/2 + O_p(1/\sqrt{n}) $ by central limit theorem and delta method, $\hat U(w;(1/2,1/2))$ converges to its limit with rate $1/\sqrt{n}$. Hence $|\hat U_{\text{ws}}(w;(1/2,1/2)) - \hat U(w;(1/2,1/2))|$ is ignorable compared to the random fluctuation of $\hat U(w;(1/2,1/2))$ when $n$ is large.

This result on convergence rate also holds when $E(Y|X)$ is linear and $\hat Y_t^\ell$ is trained with an ordinary least squares (OLS) regression. Consider $K$ studies, $$y_{i,k} = \beta_k^\intercal x_{i,k} + \epsilon_{i,k},
\label{eq:diff-sim}$$ where $\beta_k$ is a study-specific regression coefficient and $\epsilon_{i,k}$ are $N(0,\sigma^2)$ noise terms. The $x_{i,k}\sim N(0,I)$ are *iid* $p$-dimensional covariate vectors in all $K$ studies. In the following proposition, we show the respective rates at which $|\hat U_{\text{ws}}(w;1/K{\mathbf{1}}_K)-\hat U(w;1/K{\mathbf{1}}_K)|$ and $|\hat U(w;1/K{\mathbf{1}}_K) - \lim_{n\to\infty}\hat U(w;1/K{\mathbf{1}}_K)|$ converge to $0$.

\[prop:cv-eq\] Assume $\mathcal D$ is study-specific and $n_k=n$ for $k\in\{1,2,\ldots,K\}$, where $n$ is divisible by $M$. Fix $\beta_1,\ldots,\beta_K$ and assume the data are generated with (\[eq:diff-sim\]). Let $L=1$ and the single learner be an OLS procedure. If any sub-matrix $X_k'$ formed by $(1-1/M)n$ rows of $X_k$ is invertible 