\begin{align*}
    \text{Cov}&[H^{0i}_{\tau,k(j)}, H^{0i}_{\tau,k'(j)}]
    =f^{0i}_{*,\tau}(\min(k,k')) \bigg{\{}\\
    &-U_{\tau}(\max(k,k'))
    \Pr(X_j \leq C_j, Z_{X_j} = i, 
    \min(X_j, C_j) = \max(k,k'), 
    Y_j \leq \min(k,k'))\\
    &+ f^{0i}_{*,\tau}(\max(k,k'))
    \Pr(Y_j \leq k \leq \min(X_j, C_j), Y_j \leq k' \leq \min(X_j,C_j))
    \bigg{ \} }.
\end{align*}
