The mixture prior places local KL and log-square mass near a compact smooth truth

Since $G$ is a probability on $(0,\mathrm{\infty })={\bigcup }_{m\ge 1}[1\mathrm{/}m,m]$, choose $0<s_{-}<s_{+}<\mathrm{\infty }$ with $g:=G([s_{-},s_{+}])>0$. Put ${\sigma }_{-}=s_{-}{\sigma }_n$. Apply the moment-matching proof of [F2] to the law $P_0$ at scale ${\sigma }_{-}$, restricting each compact moment curve to $K$ intersected with the closure of its partition interval. The same convex-hull reduction and remainder bounds then give a finite discrete probability $F_n$ supported on $K$ with at most $C{\sigma }_n^{-1}\log n$ atoms and $\mathrm{\parallel }{p}_{F_n,{\sigma }_{-}}-p_{P_0,{\sigma }_{-}}{\mathrm{\parallel }}_1\le {\eta }_n$, where ${\eta }_n:={\sigma }_n^8$.

Choose an open neighborhood $V$ of $K$ on which the density of $\alpha$ is bounded above and below by positive constants. By compactness, choose a fixed finite union $D$ of disjoint compact intervals with $K\subset \mathrm{int}D\subset D\subset V$. Move each atom of $F_n$ to a nearest point of a lattice of mesh ${\eta }_n{\sigma }_{-}$, merge coincident atoms, and call the resulting law ${\tilde{F}}_n$. Each atom moves by at most ${\eta }_n{\sigma }_{-}\mathrm{/}2$; for every $\sigma \ge {\sigma }_{-}$ this changes the mixture in $L^1$ by at most ${\eta }_n\mathrm{/}2$, since $\mathrm{\parallel }{\mathrm{\partial }}_z{\varphi }_{\sigma }(\cdot -z){\mathrm{\parallel }}_1=E\mathrm{\mid }Z\mathrm{\mid }\mathrm{/}\sigma \le 1\mathrm{/}\sigma \le 1\mathrm{/}{\sigma }_{-}$, using $2\mathrm{\mid }Z\mathrm{\mid }\le 1+Z^2$ and [F8]. The relocated locations are separated by at least ${\eta }_n{\sigma }_{-}$. Around each take a centered cell of length ${\mathrm{\ell }}_n={\eta }_n{\sigma }_{-}\mathrm{/}2$. In each component of $D$, split every intervening or end gap of length $g\ge {\mathrm{\ell }}_n$ into $q=\lceil 2g\mathrm{/}{\sigma }_{-}\rceil$ equal subintervals. For large $n$, the end gaps have length at least ${\mathrm{\ell }}_n$ because $K\subset \mathrm{int}D$, and gaps between atom-cells have length at least ${\mathrm{\ell }}_n$ by lattice separation. If $q=1$, the subinterval length lies in $[{\mathrm{\ell }}_n,{\sigma }_{-}\mathrm{/}2]$; if $q\ge 2$, then $g>{\sigma }_{-}\mathrm{/}2$ and $q\le 4g\mathrm{/}{\sigma }_{-}$, so each subinterval has length in $[{\sigma }_{-}\mathrm{/}4,{\sigma }_{-}\mathrm{/}2]$. The resulting Borel partition $U_1,\dots ,U_N$ of $D$ has every cell length between $c{\eta }_n{\sigma }_{-}$ and ${\sigma }_{-}\mathrm{/}2$, every atom-cell diameter at most $2{\eta }_n{\sigma }_{-}$, and $N\le C(m_n+{\sigma }_{-}^{-1})\le C{\sigma }_n^{-1}\log n$, where $m_n$ is the number of atoms. Let $w_j={\tilde{F}}_n(U_j)$, so ${\sum }_j{w}_j=1$. For all large $n$, $c{\eta }_n{\sigma }_n\le \alpha (U_j)\le 1$ and $c{\eta }_n{\sigma }_n\ge c^{\mathrm{\prime }}{\eta }_n^2$. Also $N{\eta }_n\le C{\sigma }_n^7\log n\to 0$. The residual cell is $U_0=\mathbb{R}\setminus D$, so $\alpha (U_0)$ is fixed, possibly zero. Apply [F3] with $a=c^{\mathrm{\prime }}$, $b=2$, and $\eta ={\eta }_n$; its constant may depend on this fixed residual mass. We obtain an event $E_n$ with ${\mathrm{\Pi }}_{DP}(E_n)\ge e^{-CN\log (e\mathrm{/}{\eta }_n)}\ge e^{-C{\sigma }_n^{-1}(\log n{)}^2}$, on which $F(U_0)\le 3{\eta }_n$, ${\sum }_j\mathrm{\mid }F(U_j)-w_j\mathrm{\mid }\le 3{\eta }_n$, and $F(U_j)\ge {\eta }_n^2\mathrm{/}2$ for every $j$.

For $F$ in $E_n$, compare $p_{F,\sigma }$ with $p_{{\tilde{F}}_n,\sigma }$, uniformly for $s_{-}{\sigma }_n\le \sigma \le s_{+}{\sigma }_n$. On atom cells, replacing $F{\mathrm{\mid }}_{U_j}$ by its mass at the center costs at most $C{\eta }_n$ in $L^1$ by the [F7] location derivative bound and cell diameter. The event bounds make the total mass on gap cells and $U_0$ at most $C{\eta }_n$ and the sum of weight discrepancies at most $C{\eta }_n$. Since each normal density has $L^1$ norm one, $\mathrm{\parallel }{p}_{F,\sigma }-p_{{\tilde{F}}_n,\sigma }{\mathrm{\parallel }}_1\le C{\eta }_n$. For every such $\sigma$, the relocation estimate in step 1.2 also gives $\mathrm{\parallel }{p}_{{\tilde{F}}_n,\sigma }-p_{F_n,\sigma }{\mathrm{\parallel }}_1\le C{\eta }_n$. If $\sigma >{\sigma }_{-}$, completing the square gives ${\varphi }_{{\sigma }_{-}}\ast {\varphi }_v={\varphi }_{\sigma }$ for $v=\sqrt{{\sigma }^2-{\sigma }_{-}^2}$; at $\sigma ={\sigma }_{-}$ the identity is immediate. Convolution by a probability density contracts $L^1$ by Tonelli [F9], so $\mathrm{\parallel }{p}_{F_n,\sigma }-p_{P_0,\sigma }{\mathrm{\parallel }}_1\le \mathrm{\parallel }{p}_{F_n,{\sigma }_{-}}-p_{P_0,{\sigma }_{-}}{\mathrm{\parallel }}_1\le {\eta }_n$. Thus [F6] gives $h(p_{F,\sigma },p_{P_0,\sigma })\le C\sqrt{{\eta }_n}$, and the assumed bias bound gives $h(p_0,p_{F,\sigma })\le C({\sigma }_n^2+\sqrt{{\eta }_n})\le C{\sigma }_n^2$.

For every $x\in K$, the interval $[x-{\sigma }_{-},x+{\sigma }_{-}]$ contains a full cell $U_j$ because $D$ contains a fixed neighborhood of $K$ and all cells have length at most ${\sigma }_{-}\mathrm{/}2$. On this interval ${\varphi }_{\sigma }(x-z)\ge c\mathrm{/}\sigma$ for $s_{-}{\sigma }_n\le \sigma \le s_{+}{\sigma }_n$. Thus [F1] and the cell-mass lower bound in step 1.2 give $p_{F,\sigma }(x)\ge c{\eta }_n^2\mathrm{/}{\sigma }_n$. Since $p_0$ is bounded and vanishes outside $K$, $R_n:=\mathrm{\parallel }{p}_0\mathrm{/}{p}_{F,\sigma }{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{\sigma }_n{\eta }_n^{-2}=C{\sigma }_n^{-15}$. Step 2.1 gives $h(p_0,p_{F,\sigma })\le C{\sigma }_n^2$, and the polynomial lower bound on ${\sigma }_n$ gives $1+{\log }^{+}{R}_n\le C\log n$. Apply [F4] to $p=p_0$ and $q=p_{F,\sigma }$ to obtain both local KL quantities at most $C{\sigma }_n^4(\log n{)}^2\le C{\delta }_n^2$.

The scale event $S\in [s_{-},s_{+}]$ has probability $g>0$ and is independent of $F$. Thus the subset $E_n\times \{S\in [s_{-},s_{+}]\}$ is contained in the displayed local KL/log-square set and has prior probability at least $g{e}^{-C{\sigma }_n^{-1}(\log n{)}^2}$. By the definition of ${\delta }_n$, $n{\delta }_n^2\ge {\sigma }_n^{-1}(\log n{)}^2$, so increasing $C_1$ absorbs $g$ and proves the required lower bound.