Compact-uniform LAN and rate contraction imply a total-variation BvM limit

Let $Q_n$ be the posterior law of $H_n$. The contraction premise implies that $(Q_n)$ is tight in probability: for every $\eta ,\epsilon >0$ there is a finite $R$ such that ${\mathrm{lim\hspace{0.17em}sup}}_n{P}_{0,n}\{Q_n(\mathrm{\parallel }h{\mathrm{\parallel }}_2>R)>\eta \}<\epsilon$. Otherwise there would be $\eta ,\epsilon >0$ and increasing radii $R_k\to \mathrm{\infty }$ with indices $n_k$ for which $P_{0,n_k}\{Q_{n_k}(\mathrm{\parallel }h{\mathrm{\parallel }}_2>R_k)>\eta \}>\epsilon$. Define $M_n=R_k$ for $n_k\le n<n_{k+1}$, with any positive initial values. Then $M_n\to \mathrm{\infty }$ and the assumed contraction fails along $n_k$, a contradiction.

The supplied common-dominator density setup [F1, F2] makes the posterior likelihood ratio well-defined on the stated $P_{0,n}$-full set. Fix $R>0$ and put $K_R=\{h:\mathrm{\parallel }h{\mathrm{\parallel }}_2\le R\}$. Since ${\theta }^{\ast }$ is interior and ${\delta }_n\to 0$, ${\theta }^{\ast }+{\delta }_n{K}_R\subset \mathrm{\Theta }$ for all sufficiently large $n$. Relative to Lebesgue measure on $K_R$, the unnormalized posterior density for $H_n$ is proportional to $q_n(h)=\exp ({\mathrm{\ell }}_n({\theta }^{\ast }+{\delta }_{n}h))\pi ({\theta }^{\ast }+{\delta }_{n}h)$; the Jacobian ${\delta }_n^d$ cancels on conditioning on $K_R$. Set $g_n(h)=\exp (h^T{\mathrm{\Delta }}_n-\frac{1}{2}{h}^{T}Vh)$. By compact-uniform LAN and continuity of $\pi$ at ${\theta }^{\ast }$, $q_n(h)\mathrm{/}{g}_n(h)=\pi ({\theta }^{\ast })(1+o_{P_{0,n}}(1))$ uniformly on $K_R$. Completing the square gives $g_n$ proportional to the density in [F5] with mean $V^{-1}{\mathrm{\Delta }}_n$ and covariance $V^{-1}$. If $w_n=q_n\mathrm{/}{g}_n$ and ${\bar{w}}_n={\int }_{K_R}{w}_{n}d{G}_n^{K_R}$, where $G_n^{K_R}$ is the Gaussian law conditioned on $K_R$, the two conditional densities have ratio $w_n\mathrm{/}{\bar{w}}_n$. Uniform convergence $w_n\mathrm{/}\pi ({\theta }^{\ast })\to 1$ implies ${\bar{w}}_n\mathrm{/}\pi ({\theta }^{\ast })\to 1$, so ${\sup }_{K_R}\mathrm{\mid }{w}_n\mathrm{/}{\bar{w}}_n-1\mathrm{\mid }=o_{P_{0,n}}(1)$ and their total-variation distance tends to zero.

By [F6] and the assumed ${\mathrm{\Delta }}_n=O_{P_{0,n}}(1)$, the Gaussian means $V^{-1}{\mathrm{\Delta }}_n$ are bounded in probability. For any fixed bound on the mean, its Gaussian probability outside $K_R$ is at most the centered $N_d(0,V^{-1})$ probability outside a ball whose radius tends to infinity with $R$. For integers $m\ge 1$, the sets $F_m=\{z:\mathrm{\parallel }z{\mathrm{\parallel }}_2>m\}$ decrease to the empty set, and $F_m={⨆}_{k\ge m}(F_k\setminus F_{k+1})$. Countable additivity makes $P(F_m)$ the tail of a convergent nonnegative series, so it tends to zero. Hence the Gaussian tails are tight in probability as $R\to \mathrm{\infty }$.

For any two probability measures $P,Q$ and measurable $K$, their restrictions conditioned on $K$ satisfy $\mathrm{\parallel }P-Q{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le \mathrm{\parallel }P(\cdot \mid K)-Q(\cdot \mid K){\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}+2P(K^c)+2Q(K^c)$ whenever both conditioning probabilities are positive. Indeed, for any event $A$, split it into $A\cap K$ and $A\cap K^c$, compare the two conditional masses on $K$, and use $\mathrm{\mid }P(K)-Q(K)\mathrm{\mid }\le P(K^c)+Q(K^c)$. Choose $R$ so the posterior tail from step 1.1 and Gaussian tail from step 1.3 are arbitrarily small in probability; the conditional distance tends to zero for this fixed $R$ by step 1.2. Letting the tail tolerance decrease to zero proves the asserted full total-variation convergence under [F3].