[proofplan] Set $H_n:=r_n(X_n-x)$, so the hypothesis is $H_n\xrightarrow{d}X$ in $D$. The rescaled maps $h\mapsto r_n(\Phi(x+h/r_n)-\Phi(x))$ converge deterministically along every sequence $h_n\to h\in D_0$ by tangential Hadamard differentiability. The extended [continuous mapping theorem](/theorems/1847) then transfers the [weak convergence](/page/Weak%20Convergence) of $H_n$ to the weak convergence of the transformed variables. Finally, continuity of $\Phi'_x$ on the separable normed space $D_0$ gives the required measurability of the limit random element. [/proofplan]

[step:Define the rescaled random elements and deterministic maps] For each $n\in\mathbb N$, define the $D$-valued random element \begin{align*} H_n:(\Omega,\mathcal A,\mathbb P)\to(D,\mathcal B(D)),\qquad H_n:=r_n(X_n-x). \end{align*} The map $y\mapsto r_n(y-x)$ from $D$ to $D$ is continuous, so $H_n$ is Borel measurable. By hypothesis, \begin{align*} H_n\xrightarrow{d}X \end{align*} in $D$. Define the admissible set \begin{align*} A_n:=\{h\in D:x+r_n^{-1}h\in D_\Phi\}. \end{align*} Since $\mathbb P(X_n\in D_\Phi)=1$ and $X_n=x+r_n^{-1}H_n$, we have \begin{align*} \mathbb P(H_n\in A_n)=1. \end{align*} For each $n\in\mathbb N$, define \begin{align*} \Psi_n:A_n\to E,\qquad \Psi_n(h):=r_n\bigl(\Phi(x+r_n^{-1}h)-\Phi(x)\bigr). \end{align*} Then, on the event $\{X_n\in D_\Phi\}$, \begin{align*} \Psi_n(H_n)=r_n\bigl(\Phi(X_n)-\Phi(x)\bigr). \end{align*} Thus it suffices to prove \begin{align*} \Psi_n(H_n)\xrightarrow{d}\Phi'_x(X) \end{align*} in $E$. [/step]

[step:Verify the deterministic convergence required by the extended continuous mapping theorem]Let $(h_n)_{n\ge 1}$ be a sequence in $D$ and let $h\in D_0$ satisfy \begin{align*} h_n\to h \end{align*} in $D$, with $h_n\in A_n$ for every $n\in\mathbb N$. Define \begin{align*} t_n:=r_n^{-1}. \end{align*} Then $t_n>0$, $t_n\to 0$, and the condition $h_n\in A_n$ says precisely that \begin{align*} x+t_nh_n\in D_\Phi. \end{align*} By Hadamard differentiability of $\Phi$ at $x$ tangentially to $D_0$, \begin{align*} \Psi_n(h_n) =\frac{\Phi(x+t_nh_n)-\Phi(x)}{t_n} \to \Phi'_x(h) \end{align*} in $E$.[/step]

[guided]The purpose of this step is to check the deterministic hypothesis behind the [delta method](/theorems/1861). The random convergence $H_n\xrightarrow{d}X$ alone is not enough; we must know that the functions applied to $H_n$ behave continuously in the limiting directions. Take any sequence $(h_n)_{n\ge 1}$ in $D$ and any $h\in D_0$ such that \begin{align*} h_n\to h \end{align*} in the norm of $D$. Assume also that $h_n\in A_n$ for every $n\in\mathbb N$. By the definition \begin{align*} A_n:=\{h\in D:x+r_n^{-1}h\in D_\Phi\}, \end{align*} this means \begin{align*} x+r_n^{-1}h_n\in D_\Phi \end{align*} for every $n\in\mathbb N$. Now define \begin{align*} t_n:=r_n^{-1}. \end{align*} Because each $r_n$ is positive and $r_n\to\infty$, the sequence $(t_n)_{n\ge 1}$ lies in $(0,\infty)$ and satisfies \begin{align*} t_n\to 0. \end{align*} We are therefore exactly in the situation required by tangential Hadamard differentiability: the directions $h_n$ converge in $D$ to a direction $h\in D_0$, the perturbation sizes $t_n$ tend to $0$, and the perturbed points $x+t_nh_n$ remain in the domain $D_\Phi$. Applying the definition of Hadamard differentiability tangentially to $D_0$ gives \begin{align*} \frac{\Phi(x+t_nh_n)-\Phi(x)}{t_n}\to \Phi'_x(h) \end{align*} in $E$. Since $t_n=r_n^{-1}$, the left-hand side is exactly \begin{align*} r_n\bigl(\Phi(x+r_n^{-1}h_n)-\Phi(x)\bigr)=\Psi_n(h_n). \end{align*} Hence \begin{align*} \Psi_n(h_n)\to \Phi'_x(h) \end{align*} in $E$. This is the deterministic continuity property that replaces ordinary continuity of $\Phi$ at $x$; it is precisely where the tangential Hadamard differentiability hypothesis is used.[/guided]

[step:Apply the extended continuous mapping theorem to the rescaled maps] We use the standard extended continuous mapping theorem for varying maps on metric spaces: if $Z_n\xrightarrow{d}Z$ in a [metric space](/page/Metric%20Space), $Z$ takes values in a set $S$, and maps $g_n$ satisfy $g_n(z_n)\to g(z)$ whenever $z_n\to z\in S$, then $g_n(Z_n)\xrightarrow{d}g(Z)$, provided the displayed random variables are measurable. This is the usual extended continuous mapping theorem used in the Banach-space delta method (citing a result not yet in the wiki: Extended Continuous Mapping Theorem). Apply this theorem with the metric space $D$, the target metric space $E$, the random elements $Z_n:=H_n$, the limit $Z:=X$, the set $S:=D_0$, the maps $g_n:=\Psi_n$ on $A_n$, and the limiting map \begin{align*} g:D_0\to E,\qquad g(h):=\Phi'_x(h). \end{align*} The convergence $H_n\xrightarrow{d}X$ in $D$ is the hypothesis already established, the limit $X$ takes values in $D_0$ by assumption, and the sequential convergence condition for $g_n$ was verified in the preceding step. Since $\Psi_n(H_n)=r_n(\Phi(X_n)-\Phi(x))$ almost surely and these variables are measurable by hypothesis on $\Phi(X_n)$, the theorem yields \begin{align*} r_n\bigl(\Phi(X_n)-\Phi(x)\bigr)\xrightarrow{d}\Phi'_x(X) \end{align*} in $E$, once $\Phi'_x(X)$ is known to be an $E$-valued Borel random element. [/step]

[step:Verify measurability of the limiting random element] The derivative \begin{align*} \Phi'_x:D_0\to E \end{align*} is continuous when $D_0$ is equipped with the norm inherited from $D$. Hence $\Phi'_x$ is Borel measurable from $(D_0,\mathcal B(D_0))$ to $(E,\mathcal B(E))$. Since \begin{align*} X:(\Omega,\mathcal A,\mathbb P)\to(D_0,\mathcal B(D_0)) \end{align*} is Borel measurable by hypothesis, the composition \begin{align*} \Phi'_x\circ X:(\Omega,\mathcal A,\mathbb P)\to(E,\mathcal B(E)) \end{align*} is Borel measurable. Therefore $\Phi'_x(X)$ is an $E$-valued random element. Combining this measurability conclusion with the convergence obtained from the extended continuous mapping theorem gives \begin{align*} r_n\bigl(\Phi(X_n)-\Phi(x)\bigr)\xrightarrow{d}\Phi'_x(X) \end{align*} in the [Banach space](/page/Banach%20Space) $E$. This is the asserted Banach-space delta method. [/step]