Convolution bound for regular estimators

Under $P_{n,0}$, $T_n\Rightarrow W$ by regularity at $h=0$ and ${\mathrm{\Delta }}_n\Rightarrow N_d(0,K)$ by LAN. By [F4], both marginal sequences are tight. Their joint laws are tight: for any $ϵ>0$, choose compact balls $C_T,C_{\mathrm{\Delta }}$ with marginal outside probabilities below $ϵ\mathrm{/}2$; the union bound puts the joint law on the compact product with probability at least $1-ϵ$.

By [F4] and Prokhorov's theorem, select a subsequence along which $(T_n,{\mathrm{\Delta }}_n)\Rightarrow (T,\mathrm{\Delta })$. Its marginals are $T\sim W$ and $\mathrm{\Delta }\sim N_d(0,K)$, since the full marginal sequences already converge. This supplies the joint-limit subsequence used below.

Fix $h\in {\mathbb{R}}^d$. Along this subsequence [F1] and [F5] give $(T_n,{\mathrm{\Delta }}_n,\log L_{n,h})\Rightarrow (T,\mathrm{\Delta },h^T\mathrm{\Delta }-\frac{1}{2}{h}^{T}Kh)\mathrm{.}$ The final coordinate is normal with mean $-h^{T}Kh\mathrm{/}2$ and variance $h^{T}Kh$. By [F1], $P_{n,0}(L_{n,h}=0)\to 0$. Assign any finite value to the log on that vanishing event; [F10] then gives $(T_n,{\mathrm{\Delta }}_n,L_{n,h})\Rightarrow (T,\mathrm{\Delta },e^{h^T\mathrm{\Delta }-h^{T}Kh\mathrm{/}2})$. By [F3], under the true $P_{n,h}$ law, the first coordinate has limit $T^{(h)}$ given by tilting the joint law of $(T,\mathrm{\Delta })$ by $e^{h^T\mathrm{\Delta }-h^{T}Kh\mathrm{/}2}$. Regularity [F2] also gives $T_n\Rightarrow W+h$ under $P_{n,h}$. Therefore for every $t\in {\mathbb{R}}^d$ and every real $h$, $E\left[e^{h^T\mathrm{\Delta }-h^{T}Kh\mathrm{/}2}{e}^{i{t}^{T}T}\right]=e^{i{t}^{T}h}{\phi }_W(t),$ where ${\phi }_W$ is the characteristic function of $W$.

Rearranging step 3.1 gives, for real $h$, $F_t(h):=E[e^{i{t}^{T}T+h^T\mathrm{\Delta }}]=e^{h^{T}Kh\mathrm{/}2+i{t}^{T}h}{\phi }_W(t)\mathrm{.}$ For complex $z\in {\mathbb{C}}^d$, $F_t(z)=E[e^{i{t}^{T}T+z^T\mathrm{\Delta }}]$ is entire in each coordinate: on every compact set of $z$, its exponential power series and coordinate derivatives are dominated by $e^{R\mathrm{\parallel }\mathrm{\Delta }{\mathrm{\parallel }}_1}$ for some finite $R$, whose expectation is finite because each Gaussian coordinate has finite exponential moments. The right-hand side is also entire. Apply [F7] one coordinate at a time, holding the other coordinates real and then successively allowing them to be complex. The equality therefore holds for every complex $z$.

Set $z=iu$ in step 4.1. Then $E[e^{i{t}^{T}T+i{u}^T\mathrm{\Delta }}]={\phi }_W(t)e^{-t^{T}u-u^{T}Ku\mathrm{/}2}\mathrm{.}$ Let $R=T-K^{-1}\mathrm{\Delta }$. Replacing $u$ by $u-K^{-1}t$ gives $E[e^{i{t}^{T}R+i{u}^T\mathrm{\Delta }}]=({\phi }_W(t)e^{t^T{K}^{-1}t\mathrm{/}2})e^{-u^{T}Ku\mathrm{/}2}\mathrm{.}$ The first factor equals $E{e}^{i{t}^{T}R}$ by setting $u=0$; the second is the characteristic function of $\mathrm{\Delta }$. Thus the joint characteristic function of $(R,\mathrm{\Delta })$ factors into its marginals.

By [F12], form the product law $\rho =\mathcal{L}(R)\otimes \mathcal{L}(\mathrm{\Delta })$ on ${\mathbb{R}}^{2d}$. Step 5.1 and factorization of the bounded product integrals show that $\rho$ and $\mathcal{L}(R,\mathrm{\Delta })$ have identical joint characteristic functions. For every $a,b\in {\mathbb{R}}^d$ and real $v$, evaluate these functions at $(va,vb)$; this gives equality of the scalar characteristic functions of the projection $(r,z)\mapsto a^{T}r+b^{T}z$ under the two laws. Scalar uniqueness [F8] identifies each projection law, and Cramér–Wold [F11], under the existing AC hypothesis [A1], gives $\mathcal{L}(R,\mathrm{\Delta })=\rho$. Thus $R$ and $\mathrm{\Delta }$ are independent. Since $K$ is invertible, $K^{-1}\mathrm{\Delta }\sim N_d(0,K^{-1})$, and $T=K^{-1}\mathrm{\Delta }+R$. The law of $T$, which is $W$, is therefore $N_d(0,K^{-1})\ast \mathcal{L}(R)$, proving the claimed convolution with $\nu =\mathcal{L}(R)$.