[proofplan] The proof is the functional delta method applied twice. First, the $P_0$-Donsker and square-integrable-envelope hypotheses are the empirical-process hypotheses that justify the Brownian-bridge limits; because the statement assumes those limits explicitly, the proof uses the displayed ordinary and conditional convergence assumptions directly. Hadamard differentiability at $P_0$ then converts [weak convergence](/page/Weak%20Convergence) of the empirical process $\sqrt n(P_n-P_0)$ into weak convergence of the scalar statistic $\sqrt n(\theta(P_n)-\theta(P_0))$. Second, the bootstrap version of the same delta-method expansion is applied to the conditional perturbation $P_n^*-P_n$ given the observed data. Linearity and continuity of $\dot\theta_{P_0}$ transfer the bootstrap empirical-process limit to the same scalar limit. [/proofplan]

[step:Apply Hadamard differentiability to the empirical perturbation]Define the empirical process $Z_n:\mathcal F\to\mathbb R$ by the rule \begin{align*} Z_n(f)=\sqrt n\bigl(P_nf-P_0f\bigr) \end{align*} for each $f\in\mathcal F$. By hypothesis, \begin{align*} Z_n\xrightarrow{d}G_{P_0} \end{align*} in $\ell^\infty(\mathcal F)$. Since $T$ is a closed linear subspace of the [normed vector space](/page/Normed%20Vector%20Space) $\ell^\infty(\mathcal F)$ and $\dot\theta_{P_0}:T\to\mathbb R$ is continuous and linear, the [Hahn-Banach Theorem](/theorems/879) gives a continuous [linear map](/page/Linear%20Map) $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ whose restriction to $T$ is $\dot\theta_{P_0}$. Define the empirical Hadamard remainder $r_n$ by \begin{align*} r_n = \sqrt n\bigl(\theta(P_n)-\theta(P_0)\bigr) - \widetilde{\dot\theta}_{P_0}(Z_n). \end{align*} We invoke the functional form of the [Delta Method](/theorems/1861) for Hadamard differentiable maps, applied to the map $\theta:\mathcal P\to\mathbb R$ at $P_0$ with tangent set $T$. Its hypotheses are satisfied: $t_n=n^{-1/2}\downarrow0$, $P_n=P_0+t_nZ_n$ as elements of $\ell^\infty(\mathcal F)$ on the event $P_n\in\mathcal P$, this event has probability tending one by the theorem statement, the sequence $Z_n$ converges weakly to $G_{P_0}$ in $\ell^\infty(\mathcal F)$ by hypothesis, $G_{P_0}$ takes values in the closed tangent set $T$ almost surely, and the measurability or outer-probability convention in the statement is exactly the convention required to interpret the weak convergence. The theorem therefore gives the asymptotic linear expansion \begin{align*} r_n\to0 \end{align*} in probability. Because $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ is continuous and linear, applying the [Continuous Mapping Theorem](/theorems/1847) to \begin{align*} Z_n\xrightarrow{d}G_{P_0} \end{align*} gives \begin{align*} \widetilde{\dot\theta}_{P_0}(Z_n)\xrightarrow{d}\widetilde{\dot\theta}_{P_0}(G_{P_0})=\dot\theta_{P_0}(G_{P_0}), \end{align*} where the final equality uses $G_{P_0}\in T$ almost surely. Since $r_n\to0$ in probability, [Slutsky's Lemma](/theorems/1850) yields \begin{align*} \sqrt n\bigl(\theta(P_n)-\theta(P_0)\bigr) \xrightarrow{d} \dot\theta_{P_0}(G_{P_0}). \end{align*}[/step]

[guided]The empirical measure $P_n$ is viewed as an element of $\ell^\infty(\mathcal F)$ through the evaluation map $f\mapsto P_nf$. Thus the natural first-order fluctuation of $P_n$ around $P_0$ is the function $Z_n:\mathcal F\to\mathbb R$ defined by \begin{align*} Z_n(f)=\sqrt n\bigl(P_nf-P_0f\bigr) \end{align*} for each $f\in\mathcal F$. The hypothesis says exactly that $Z_n$ converges weakly in $\ell^\infty(\mathcal F)$ to the $P_0$-Brownian bridge $G_{P_0}$. Hadamard differentiability of $\theta$ at $P_0$ tangentially to $T$ is a deterministic first-order condition: whenever $t_n\downarrow0$ and $h_n\to h$ in $\ell^\infty(\mathcal F)$ with $h\in T$ and $P_0+t_nh_n\in\mathcal P$, one has \begin{align*} \frac{\theta(P_0+t_nh_n)-\theta(P_0)}{t_n} \to \dot\theta_{P_0}(h). \end{align*} The empirical perturbations do not converge to one fixed deterministic direction; instead $Z_n$ converges weakly to the random direction $G_{P_0}$. This is exactly why the functional delta method is needed. We apply that theorem with $t_n=n^{-1/2}$ and with $P_n=P_0+t_nZ_n$ as elements of $\ell^\infty(\mathcal F)$. Its hypotheses are satisfied because $Z_n\xrightarrow{d}G_{P_0}$ in $\ell^\infty(\mathcal F)$, the Brownian bridge $G_{P_0}$ belongs to the closed tangent set $T$ almost surely, $\theta$ is Hadamard differentiable at $P_0$ tangentially to $T$, and the theorem statement supplies the measurability or outer-probability interpretation needed for the random variables involved. Since $T$ is a closed linear subspace of $\ell^\infty(\mathcal F)$ and $\dot\theta_{P_0}:T\to\mathbb R$ is continuous and linear, the Hahn-Banach [extension theorem](/theorems/59) gives a continuous linear map $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ whose restriction to $T$ is $\dot\theta_{P_0}$. The functional delta method therefore supplies the expansion \begin{align*} \sqrt n\bigl(\theta(P_n)-\theta(P_0)\bigr) = \widetilde{\dot\theta}_{P_0}(Z_n)+r_n, \end{align*} where the real-valued [random variable](/page/Random%20Variable) $r_n$ is the empirical Hadamard remainder and $r_n\to0$ in probability. The remaining point is to pass the weak limit through the derivative extension. The map $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ is continuous and linear by hypothesis, so it preserves weak convergence under continuous mapping: \begin{align*} Z_n\xrightarrow{d}G_{P_0} \quad\Longrightarrow\quad \widetilde{\dot\theta}_{P_0}(Z_n)\xrightarrow{d}\widetilde{\dot\theta}_{P_0}(G_{P_0})=\dot\theta_{P_0}(G_{P_0}). \end{align*} Finally, the remainder $r_n$ vanishes in probability. Adding a term converging to $0$ in probability does not change the distributional limit, so \begin{align*} \sqrt n\bigl(\theta(P_n)-\theta(P_0)\bigr) \xrightarrow{d} \dot\theta_{P_0}(G_{P_0}). \end{align*}[/guided]

[step:Linearize the bootstrap statistic conditionally on the data]Define the bootstrap empirical process $Z_n^*:\mathcal F\to\mathbb R$ by the rule \begin{align*} Z_n^*(f)=\sqrt n\bigl(P_n^*f-P_nf\bigr) \end{align*} for each $f\in\mathcal F$. Here $X_1,\dots,X_n$ denote the observed sample whose empirical measure is $P_n$. By hypothesis, conditionally on the observed data $X_1,\dots,X_n$, \begin{align*} Z_n^*\xrightarrow{d}G_{P_0} \end{align*} in probability in $\ell^\infty(\mathcal F)$. Let $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ be the continuous linear extension of $\dot\theta_{P_0}$ obtained from the [Hahn-Banach Theorem](/theorems/879). Set $t_n=n^{-1/2}$. Since $Z_n\xrightarrow{d}G_{P_0}$, the sequence $(Z_n)$ is tight in $\ell^\infty(\mathcal F)$. Hence \begin{align*} \|P_n-P_0\|_{\ell^\infty(\mathcal F)} = t_n\|Z_n\|_{\ell^\infty(\mathcal F)} \to0 \end{align*} in probability. Thus the random centering points $P_n$ approach the differentiability point $P_0$. We now apply the bootstrap functional delta-method theorem, equivalently the moving-center form of [Bootstrap Consistency](/theorems/1995) combined with the functional [Delta Method](/theorems/1861), to the maps \begin{align*} a_n:=P_n, \qquad h_n^*:=Z_n^*, \qquad t_n:=n^{-1/2}. \end{align*} The hypotheses are the following and are satisfied here. First, $a_n\to P_0$ in probability as just shown. Second, on events whose probability tends to one, $a_n\in\mathcal P$, $a_n+t_nh_n^*=P_n^*\in\mathcal P$, and \begin{align*} P_n^*=P_n+t_nZ_n^* \end{align*} holds in $\ell^\infty(\mathcal F)$. Third, conditionally on the observed data, $h_n^*=Z_n^*$ converges weakly in probability in $\ell^\infty(\mathcal F)$ to the $T$-valued random element $G_{P_0}$. Fourth, $\theta$ is Hadamard differentiable at $P_0$ tangentially to the closed linear tangent set $T$, with continuous derivative $\dot\theta_{P_0}:T\to\mathbb R$. The theorem therefore gives a real-valued conditional remainder $r_n^*$ such that $r_n^*\to0$ in [conditional probability](/page/Conditional%20Probability), in probability, and \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) = \widetilde{\dot\theta}_{P_0}(Z_n^*)+r_n^*. \end{align*} This applies the bootstrap delta method at the moving center $P_n$; it does not require subtracting two uncontrolled Hadamard expansions from the fixed point $P_0$.[/step]

[guided]The bootstrap statistic is centered at $P_n$, so the perturbation that enters the delta method is the conditional fluctuation around $P_n$. Let $Z_n^*:\mathcal F\to\mathbb R$ be the function defined by \begin{align*} Z_n^*(f)=\sqrt n\bigl(P_n^*f-P_nf\bigr) \end{align*} for each $f\in\mathcal F$. This gives the identity \begin{align*} P_n^*=P_n+n^{-1/2}Z_n^* \end{align*} inside $\ell^\infty(\mathcal F)$. The point that needs care is the random centering. We should not expand both $P_n$ and $P_n^*$ from $P_0$ and then subtract unless a uniform Hadamard remainder has been proved. Instead, we use the moving-center bootstrap delta method. First verify that the moving center approaches $P_0$. Since \begin{align*} Z_n=\sqrt n(P_n-P_0) \end{align*} and $Z_n\xrightarrow{d}G_{P_0}$ in $\ell^\infty(\mathcal F)$, the sequence $(Z_n)$ is tight. Therefore, with $t_n=n^{-1/2}$, \begin{align*} \|P_n-P_0\|_{\ell^\infty(\mathcal F)} = t_n\|Z_n\|_{\ell^\infty(\mathcal F)} \to0 \end{align*} in probability. Thus $P_n$ is a valid random center converging to the differentiability point $P_0$. Let $\widetilde{\dot\theta}_{P_0}:\ell^\infty(\mathcal F)\to\mathbb R$ be the continuous linear extension of $\dot\theta_{P_0}$ obtained from the [Hahn-Banach Theorem](/theorems/879). We apply the moving-center form of [Bootstrap Consistency](/theorems/1995) together with the functional [Delta Method](/theorems/1861) to \begin{align*} a_n:=P_n, \qquad h_n^*:=Z_n^*, \qquad t_n:=n^{-1/2}. \end{align*} The theorem requires the centers $a_n$ to converge to $P_0$, the perturbed points $a_n+t_nh_n^*$ to lie in the domain with probability tending one, conditional weak convergence of $h_n^*$ to a tangent-valued limit, and Hadamard differentiability at $P_0$ tangentially to that tangent set. We have just shown $a_n=P_n\to P_0$ in probability. On events whose probability tends to one, $P_n\in\mathcal P$ and \begin{align*} a_n+t_nh_n^*=P_n+n^{-1/2}Z_n^*=P_n^*\in\mathcal P. \end{align*} By assumption, conditionally on the observed data, \begin{align*} Z_n^*\xrightarrow{d}G_{P_0} \end{align*} in probability in $\ell^\infty(\mathcal F)$, and $G_{P_0}$ takes values in the closed tangent set $T$ almost surely. Finally, $\theta$ is Hadamard differentiable at $P_0$ tangentially to $T$ with continuous derivative $\dot\theta_{P_0}:T\to\mathbb R$, and the theorem statement supplies measurability or the outer-probability convention needed for the conditional statement. The bootstrap delta method therefore gives the expansion \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) = \widetilde{\dot\theta}_{P_0}(Z_n^*)+r_n^*, \end{align*} where $r_n^*$ is a real-valued bootstrap remainder satisfying $r_n^*\to0$ in conditional probability, in probability. The important point is that the derivative is still the derivative at $P_0$, but the remainder control is obtained from a theorem designed for the moving random center $P_n$.[/guided]

[step:Pass the conditional bootstrap law through the derivative]Let $\mathrm{BL}_1(\mathbb R)$ denote the set of all functions $\varphi:\mathbb R\to\mathbb R$ such that $|\varphi(u)|\le1$ for all $u\in\mathbb R$ and \begin{align*} |\varphi(u)-\varphi(v)|\le |u-v| \end{align*} for all $u,v\in\mathbb R$. Let $\mathrm{BL}_1(\ell^\infty(\mathcal F))$ denote the set of all functions $\psi:\ell^\infty(\mathcal F)\to\mathbb R$ such that $|\psi(h)|\le1$ for all $h\in\ell^\infty(\mathcal F)$ and \begin{align*} |\psi(h)-\psi(k)|\le \|h-k\|_{\ell^\infty(\mathcal F)} \end{align*} for all $h,k\in\ell^\infty(\mathcal F)$. Let $(\Omega^*,\mathcal A^*,\mathbb P^*)$ denote the auxiliary bootstrap probability space used conditionally on the observed data $X_1,\dots,X_n$. Let $\mathbb E_n^*[\cdot]$ denote expectation with respect to the auxiliary bootstrap law conditional on the observed data $X_1,\dots,X_n$. Conditional weak convergence in probability of $Z_n^*$ to $G_{P_0}$ means that \begin{align*} \sup_{\psi\in\mathrm{BL}_1(\ell^\infty(\mathcal F))} \left| \mathbb E_n^*\!\left[\psi(Z_n^*)\right] - \mathbb E\!\left[\psi(G_{P_0})\right] \right| \to0 \end{align*} in probability; if measurability is unavailable, $\mathbb E_n^*$ and the displayed supremum are interpreted as outer [conditional expectation](/page/Conditional%20Expectation) and outer bounded-Lipschitz distance. Define the operator norm constant $L\in[0,\infty)$ by \begin{align*} L=\sup\left\{\left|\widetilde{\dot\theta}_{P_0}(h)\right|:h\in\ell^\infty(\mathcal F),\ \|h\|_{\ell^\infty(\mathcal F)}\le1\right\}. \end{align*} For each $\varphi\in\mathrm{BL}_1(\mathbb R)$, define $\psi_\varphi:\ell^\infty(\mathcal F)\to\mathbb R$ by \begin{align*} \psi_\varphi(h)=\varphi\bigl(\widetilde{\dot\theta}_{P_0}(h)\bigr) \end{align*} for $h\in\ell^\infty(\mathcal F)$. Then $|\psi_\varphi|\le1$ and \begin{align*} |\psi_\varphi(h)-\psi_\varphi(k)| \le L\|h-k\|_{\ell^\infty(\mathcal F)}. \end{align*} Let $M:=\max\{1,L\}$. If $L>0$, then $M^{-1}\psi_\varphi$ belongs to $\mathrm{BL}_1(\ell^\infty(\mathcal F))$, because its sup-norm is at most $M^{-1}\le1$ and its Lipschitz constant is at most $L/M\le1$. If $L=0$, the map $\widetilde{\dot\theta}_{P_0}$ is zero and the convergence below is immediate because $\psi_\varphi$ is constant. Therefore the conditional bounded-Lipschitz convergence of $Z_n^*$ yields \begin{align*} \sup_{\varphi\in\mathrm{BL}_1(\mathbb R)} \left| \mathbb E_n^*\!\left[\varphi\bigl(\widetilde{\dot\theta}_{P_0}(Z_n^*)\bigr)\right] - \mathbb E\!\left[\varphi\bigl(\widetilde{\dot\theta}_{P_0}(G_{P_0})\bigr)\right] \right| \le M \sup_{\psi\in\mathrm{BL}_1(\ell^\infty(\mathcal F))} \left| \mathbb E_n^*\!\left[\psi(Z_n^*)\right] - \mathbb E\!\left[\psi(G_{P_0})\right] \right| \to0 \end{align*} in probability. Since $G_{P_0}\in T$ almost surely, $\widetilde{\dot\theta}_{P_0}(G_{P_0})=\dot\theta_{P_0}(G_{P_0})$. Since $r_n^*\to0$ in conditional probability, in probability, the conditional form of [Slutsky's Lemma](/theorems/1850) permits replacing $\widetilde{\dot\theta}_{P_0}(Z_n^*)$ by \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) \end{align*} without changing the conditional weak limit. Therefore, conditionally on the observed data $X_1,\dots,X_n$, the law of \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) \end{align*} converges in probability to the law of $\dot\theta_{P_0}(G_{P_0})$.[/step]

[guided]We now prove the conditional weak convergence after applying the derivative. Let $(\Omega^*,\mathcal A^*,\mathbb P^*)$ denote the auxiliary bootstrap probability space used conditionally on the observed data. Let $\mathbb E_n^*[\cdot]$ denote expectation with respect to the auxiliary bootstrap law conditional on the observed data $X_1,\dots,X_n$. The conditional convergence of $Z_n^*$ is stated in the bounded-Lipschitz metric on $\ell^\infty(\mathcal F)$: \begin{align*} \sup_{\psi\in\mathrm{BL}_1(\ell^\infty(\mathcal F))} \left| \mathbb E_n^*\!\left[\psi(Z_n^*)\right] - \mathbb E\!\left[\psi(G_{P_0})\right] \right| \to0 \end{align*} in probability, with $\mathbb E_n^*$ interpreted as outer conditional expectation if measurability is not available. To transfer this convergence to the scalar bootstrap statistic, fix $\varphi\in\mathrm{BL}_1(\mathbb R)$ and define the map $\psi_\varphi:\ell^\infty(\mathcal F)\to\mathbb R$ by \begin{align*} \psi_\varphi(h)=\varphi\bigl(\widetilde{\dot\theta}_{P_0}(h)\bigr) \end{align*} for $h\in\ell^\infty(\mathcal F)$. Define \begin{align*} L=\sup\left\{\left|\widetilde{\dot\theta}_{P_0}(h)\right|:h\in\ell^\infty(\mathcal F),\ \|h\|_{\ell^\infty(\mathcal F)}\le1\right\} \end{align*} and set $M:=\max\{1,L\}$. Then $M^{-1}\psi_\varphi\in\mathrm{BL}_1(\ell^\infty(\mathcal F))$ whenever $L>0$, since the sup-norm is at most $M^{-1}\le1$ and the Lipschitz constant is at most $L/M\le1$. If $L=0$, then $\widetilde{\dot\theta}_{P_0}$ is the zero map and the scalar convergence is immediate. Hence the conditional bounded-Lipschitz convergence of $Z_n^*$ gives \begin{align*} \sup_{\varphi\in\mathrm{BL}_1(\mathbb R)} \left| \mathbb E_n^*\!\left[\varphi\bigl(\widetilde{\dot\theta}_{P_0}(Z_n^*)\bigr)\right] - \mathbb E\!\left[\varphi\bigl(\widetilde{\dot\theta}_{P_0}(G_{P_0})\bigr)\right] \right| \le M \sup_{\psi\in\mathrm{BL}_1(\ell^\infty(\mathcal F))} \left| \mathbb E_n^*\!\left[\psi(Z_n^*)\right] - \mathbb E\!\left[\psi(G_{P_0})\right] \right| \to0 \end{align*} in probability. Since $G_{P_0}\in T$ almost surely, the extension agrees with the original derivative on the limit: \begin{align*} \widetilde{\dot\theta}_{P_0}(G_{P_0})=\dot\theta_{P_0}(G_{P_0}). \end{align*} The previous step showed \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) = \widetilde{\dot\theta}_{P_0}(Z_n^*)+r_n^*, \end{align*} where $r_n^*\to0$ in conditional probability, in probability. The conditional form of [Slutsky's Lemma](/theorems/1850) therefore allows the vanishing conditional remainder to be removed without changing the conditional weak limit. Thus, conditionally on the observed data, the law of \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) \end{align*} converges in probability to the law of $\dot\theta_{P_0}(G_{P_0})$.[/guided]

[step:Identify the common limiting law] The first step proved \begin{align*} \sqrt n\bigl(\theta(P_n)-\theta(P_0)\bigr) \xrightarrow{d} \dot\theta_{P_0}(G_{P_0}). \end{align*} The preceding step proved that the conditional law of \begin{align*} \sqrt n\bigl(\theta(P_n^*)-\theta(P_n)\bigr) \end{align*} converges in probability to the same law, namely the law of $\widetilde{\dot\theta}_{P_0}(G_{P_0})=\dot\theta_{P_0}(G_{P_0})$. These two conclusions are exactly the asserted ordinary delta-method limit and bootstrap consistency statement. [/step]