bound on the magnitude of the reminder term in the Taylor series expansion of $g(\hat{\psi})$ around $g(\psi)$, as detailed in the proof of below. Of course, we may relax the requirement that holds almost everywhere to the requirement that it holds on an event of high probability. This is indeed the strategy we use when in applying the present results to the projection parameters in .

The covariance matrix of the linear approximation of $g(\psi)  -
g(\hat{\psi})$, which, for any $P \in
\mathcal{P}_n$, is given by $$\label{eq:Gamma}
 \Gamma  = \Gamma(\psi(P),P)=G(\psi(P)) V(P) G(\psi(P))^\top,$$ plays a crucial role in our analysis. In particular, our results will depend on the smallest variance of the linear approximation to $g(\psi)  - g(\hat{\psi})$: $$\label{eq:sigma}
    \underline{\sigma}^2  =  \inf_{ P  \in \mathcal{P}_n}\min_{j =1,\ldots,s}
    G^\top_j(\psi(P)) V(P) G_j(\psi(P)).$$

With these definitions in place we are now ready to prove the following high-dimensional Berry-Esseen bound.

\[theorem::deltamethod\] Assume that $W_1,\ldots, W_n$ is an i.i.d. sample from some $P \in {\cal P}_n$ and let $Z_n \sim N(0,\Gamma)$. Then, there exists a $C>0$, dependent on $A$ only, such that $$\sup_{P\in {\cal P}_n}
\sup_{t > 0}
\Bigl|\mathbb{P}( \sqrt{n}||\hat\theta - \theta||_\infty \leq t) -\mathbb{P}(
||Z_n||_\infty \leq t)\Bigr| \leq C \Big(
\Delta_{n,1} + \Delta_{n,2} \Big),$$ where $$\begin{aligned}
\label{eq::Delta}
\Delta_{n,1} &=
\frac{1}{\sqrt{v}} \left( \frac{ \overline{v}^2 b (\log 2bn)^7}{n}   \right)^{1/6}
\\
\Delta_{n,2} &=
\frac{1}{\underline{\sigma}}\sqrt{\frac{ b \overline{v} \overline{H}^2 (\log n)^2 \log b}{n}}.\end{aligned}$$

[**Remarks**]{} The assumption that the support of $P$ is compact is made for simplicity, and can be modified by assuming that the coordinates of the vectors $W_i$ have sub-exponential behavior. Notice also that the coordinates of the $W_i$’s need not be independent.

The proof of resembles the classic proof of the asymptotic normality of non-linear functions of ave