Interval masses give a log-free local KL neighborhood for the mixture prior

Choose $0<s_{-}<s_{+}<\mathrm{\infty }$ with $g:=G([s_{-},s_{+}])>0$, possible because these compact intervals exhaust $(0,\mathrm{\infty })$. Set $b_n={\sigma }_n$, $\tau =s_{-}{b}_n$, $B=a+6$ and $\eta =b_n^B$. For all large $n$, partition each nondegenerate component of $K$ into equal consecutive Borel intervals of lengths in $[\tau \mathrm{/}8,\tau \mathrm{/}4]$: use $\lceil 4\mathrm{\ell }\mathrm{/}\tau \rceil$ pieces for a component of length $\mathrm{\ell }$. Endpoints can be assigned arbitrarily because $P_0$ has a density. The support of a continuous density has no isolated points, so there are no isolated support components to treat. There are $O(b_n^{-1})$ coarse cells $I_k$, each of mass at least $c^{\mathrm{\prime }}{b}_n^a$. Translate the conditional probability $P_0(\cdot \mid I_k)$ to an interval centered at zero, apply [F2] at scale $\tau$, and translate back. Multiplying by $P_0(I_k)$ and summing gives a finite probability $F_n$ with at most $C{b}_n^{-1}\log (e\mathrm{/}\eta )$ atoms in the closures of these cells, each carrying its original coarse-cell label, and $\mathrm{\parallel }{p}_{F_n,\tau }-p_{P_0,\tau }{\mathrm{\parallel }}_{\mathrm{\infty }}\le \eta \mathrm{/}\tau$. The total weight of atoms with label $k$ is exactly $P_0(I_k)$.

Put $\delta =\eta \tau$. Move each atom to a nearest point of the global lattice of mesh $\delta$ and merge atoms at the same location. Preserve its fractional label weights when merging. The resulting distinct locations $z_j$ are separated by at least $\delta$, with weights $w_j$ summing to one, and still number $N\le C{b}_n^{-1}\log (e\mathrm{/}\eta )$. Let $U_j=[z_j-\delta \mathrm{/}4,z_j+\delta \mathrm{/}4)$ and $U_0=\mathbb{R}\setminus {\bigcup }_{j=1}^N{U}_j$. These are disjoint partition cells. The base density is bounded above and below on a fixed compact neighborhood of $K$, so $c_0\eta \tau \le \alpha (U_j)\le C_0\eta \tau \le 1$ eventually. Since $B>1$, $\eta \tau \ge c_1{\eta }^2$ and $N\eta \to 0$. Also ${\sum }_j\alpha (U_j)\le CN\delta \to 0$, so, writing $m=\alpha (\mathbb{R})>0$, we have $\alpha (U_0)\in [m\mathrm{/}2,m]$ eventually.

Apply [F3] with tolerance $\eta$ to get an event $E_n$ on which ${\sum }_j\mathrm{\mid }F(U_j)-w_j\mathrm{\mid }+F(U_0)\le 3\eta$ and ${\mathrm{\Pi }}_{DP}(E_n)\ge \exp \{-C{b}_n^{-1}(\log n{)}^2\}$. Its constants can be taken uniform in $n$: in the proof of [F3] the only residual-parameter factors are $\mathrm{\Gamma }(m)\mathrm{/}\mathrm{\Gamma }(\alpha (U_0))$ and a lower bound $({\eta }^2\mathrm{/}2{)}^m$ for the residual power. The gamma ratio is bounded below on the fixed compact interval $[m\mathrm{/}2,m]$, and the latter power already has fixed exponent $m$. The remaining constants use only the fixed bound $c_1{\eta }^2\le \alpha (U_j)\le 1$. Finally $\log (e\mathrm{/}\eta )\le C\log n$ by the polynomial lower bandwidth bound.

For $\tau \le \sigma \le s_{+}{b}_n$, the original discrete approximation still satisfies $\mathrm{\parallel }{p}_{F_n,\sigma }-p_{P_0,\sigma }{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\eta \mathrm{/}{b}_n$. Indeed the Gaussian convolution identity ${\varphi }_u\ast {\varphi }_v={\varphi }_{\sqrt{u^2+v^2}}$ follows by completing the square and integrating; convolution with a probability kernel contracts the supremum norm by the triangle inequality. Use $u=\tau$, $v=\sqrt{{\sigma }^2-{\tau }^2}$, treating equality by the original approximation. Differentiating [F5] gives $\mathrm{\parallel }{\varphi }_{\sigma }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{\sigma }^{-2}$. Atom relocation by at most $\delta \mathrm{/}2$ therefore changes the mixture by at most $C\delta \mathrm{/}{\sigma }^2\le C\eta \mathrm{/}{b}_n$. For $F\in E_n$, replacing each $F{\mathrm{\mid }}_{U_j}$ by its mass at $z_j$ costs at most the same derivative bound times $\delta \mathrm{/}4$; the weight discrepancies and residual mass cost at most $3\eta \mathrm{\parallel }{\varphi }_{\sigma }{\mathrm{\parallel }}_{\mathrm{\infty }}$. Consequently, uniformly over $F\in E_n$ and $\tau \le \sigma \le s_{+}{b}_n$, one has $\mathrm{\parallel }{p}_{F,\sigma }-p_{P_0,\sigma }{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\eta \mathrm{/}{b}_n$.

For any $x\in K$ choose a coarse cell whose closure contains $x$. Its length is at most $\tau \mathrm{/}4$, so every original atom with that label, and every associated relocated atom-cell, lies in $[x-\tau ,x+\tau ]$ eventually. Let $J_x$ be the set of relocated atom indices carrying a positive fraction of this label's weight. Merging preserves the full label weight, so ${\sum }_{j\in J_x}{w}_j\ge P_0(I_k)\ge c^{\mathrm{\prime }}{b}_n^a$. The vector discrepancy on $E_n$ implies $F([x-\tau ,x+\tau ])\ge {\sum }_{j\in J_x}F(U_j)\ge c^{\mathrm{\prime }}{b}_n^a-3\eta \ge c^{\mathrm{\prime }}{b}_n^a\mathrm{/}2$. On this neighborhood, [F5] is at least $c^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}{b}_n$ for every $\tau \le \sigma \le s_{+}{b}_n$. Thus $p_{F,\sigma }(x)\ge c^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}{b}_n^{a-1}$. The same argument directly with the coarse-cell mass under $P_0$ gives $p_{P_0,\sigma }(x)\ge c^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}{b}_n^{a-1}$, after decreasing the constant.

Put $q=p_{P_0,\sigma }$ and $r=p_{F,\sigma }$. The mean-value theorem for the logarithm, whose derivative is given by [F6], and steps 4.1, 4.2 give ${\sup }_{x\in K}\mathrm{\mid }\log (q(x)\mathrm{/}r(x))\mathrm{\mid }\le C\eta b_n^{-a}=C{b}_n^6$. By [F1], $\mathrm{K}\mathrm{L}(P_0\mathrm{\parallel }Q)\le C{b}_n^4$ and $P_0[\log (p_0\mathrm{/}q){]}^2\le C{b}_n^4$ uniformly in these scales. Adding the bounded log perturbation proves $\mathrm{K}\mathrm{L}(P_0\mathrm{\parallel }{P}_{F,\sigma })\le C{b}_n^4+C{b}_n^6\le C^{\mathrm{\prime }}{b}_n^4$. The inequality $(u+v{)}^2\le 2u^2+2v^2$ similarly gives $P_0[\log (p_0\mathrm{/}r){]}^2\le C^{\mathrm{\prime }}{b}_n^4$. All integrals exist by the displayed bounds, and integrands vanish off $\{p_0>0\}$.

The scale event $S\in [s_{-},s_{+}]$ has fixed probability $g>0$ and is independent of $F$. Therefore $E_n$ times this event is contained in the displayed KL/log-square set and has probability at least $g\exp \{-C{b}_n^{-1}(\log n{)}^2\}$. Increase $C_2$ to absorb $g$. Since $n{d}_n^2\ge b_n^{-1}(\log n{)}^2$ and $d_n^2\ge b_n^4$, the consequent local-radius form follows as well. This proves all assertions without a unimodality hypothesis.