Posterior quantiles stabilize under scalar weak convergence

Fix $x\in \mathbb{R}$ and $\eta >0$. Choose $0<\delta <\min (1,\eta )$. The piecewise-linear ramps that fall from $1$ to $0$ on $[x-\delta ,x]$ and on $[x,x+\delta ]$ bound $1_{(-\mathrm{\infty },x]}$ below and above. Multiplying each ramp by $\delta$ makes its sup norm and Lipschitz constant at most $1$. Comparing the lower and upper ramps under $Q_n$ with their integrals under $Q$ gives $\mathrm{\mid }{Q}_n((-\mathrm{\infty },x])-F(x)\mathrm{\mid }\le {\delta }^{-1}{\beta }_{BL}(Q_n,Q)+Q([x-\delta ,x+\delta ])$. By [A1], $Q(\{x\})=0$; [F2] then gives $Q([x-\delta ,x+\delta ])\to 0$ as $\delta \downarrow 0$. First choose $\delta$ small and then let $n\to \mathrm{\infty }$; this proves $Q_n((-\mathrm{\infty },x])\to F(x)$ in probability.

Write $q=q_u(Q)$ as in [F1]. Since [A1] makes $F$ continuous, the CDF bracket in [F1] gives $F(q)=u$. For every sufficiently small $\epsilon >0$, strict increase near $q$ then gives $F(q-\epsilon )<u<F(q+\epsilon )$. Step 1.1 at these two points implies, with probability tending to one, $Q_n((-\mathrm{\infty },q-\epsilon ])<u<Q_n((-\mathrm{\infty },q+\epsilon ])$. By monotonicity, no rational at or below $q-\epsilon$ is in the defining set for $q_u(Q_n)$, while every rational above $q+\epsilon$ belongs to it. Its infimum therefore lies in $[q-\epsilon ,q+\epsilon ]$. Since arbitrarily small such $\epsilon$ are available, the quantiles converge in probability.