The sample mean attains the unrestricted mean bound

For each bounded $s$ with $Ps=0$, choose $t\ge 0$ so small that $\mathrm{\mid }ts\mathrm{\mid }\le 1\mathrm{/}2$ everywhere. On $[-1\mathrm{/}2,1\mathrm{/}2]$, $g(x)=\sqrt{1+x}$ has $g^{\mathrm{\prime }\mathrm{\prime }}(x)=-1\mathrm{/}(4(1+x{)}^{3\mathrm{/}2})$, so $\mathrm{\mid }{g}^{\mathrm{\prime }\mathrm{\prime }}(x)\mathrm{\mid }\le 1$. [F12] with $n=1$ gives $\sqrt{1+ts}=1+\frac{t}{2}s+O(t^2\mathrm{\parallel }s{\mathrm{\parallel }}_{\mathrm{\infty }}^2)$ uniformly. The remainder is $o(t)$ in $L^2(P)$, so this is a DQM path with score $s$. Its mean is $\psi (P_{t,s})=\mu +tP[Xs]$, hence ${\dot{\psi }}_P(s)=P[(X-\mu )s]$.

Bounded centered functions are dense in $L_0^2(P)$: for $g\in L_0^2(P)$, the bounded functions $g_M=g{\mathbf{1}}_{\{\mathrm{\mid }g\mathrm{\mid }\le M\}}-P[g{\mathbf{1}}_{\{\mathrm{\mid }g\mathrm{\mid }\le M\}}]$ are centered and converge to $g$ in $L^2(P)$ by truncation and $Pg=0$. Thus $\mathcal{T}=L_0^2(P)$. By [F10], ${\dot{\psi }}_P$ is continuous and is represented by $\varphi =X-\mu \in \mathcal{T}$; [F9] makes this the canonical gradient. Its covariance is $V_{\mathrm{c}\mathrm{a}\mathrm{n}}=P[(X-\mu {)}^2]={\sigma }^2$.

The identity ${\bar{X}}_n-\mu =(P_n-P)(X-\mu )$ proves exact asymptotic linearity with influence $\phi =X-\mu$. Therefore [F7] gives the base limit $N(0,{\sigma }^2)$.

Fix a bounded centered score $s$ and $h\ge 0$. Under the base law, the iid vectors $(X_i-\mu ,s(X_i))$ have finite second moment. By [F4], their normalized sums converge jointly to a centered Gaussian with covariance entries ${\sigma }^2$, $P[(X-\mu )s]$, and $P[s^2]$. The lemma [F5] also gives $P(L_n=0)\to 0$; set ${\mathrm{\ell }}_n=\log L_n$ when $L_n>0$ and ${\mathrm{\ell }}_n=0$ otherwise, which changes the joint law only on an $o(1)$-probability event. Scalar Slutsky and Cramér–Wold [F11] give the joint Gaussian limit with ${\mathrm{\ell }}_n$; exponentiation gives the likelihood-ratio limit required by [F6]. The proxy limit of $\sqrt{n}({\bar{X}}_n-\mu )$ is therefore $N(hP[(X-\mu )s],{\sigma }^2)$, and total-variation closeness [F5] transfers it to the actual local law.

The target shift is exact: $\sqrt{n}\{\psi (P_{h\mathrm{/}\sqrt{n},s})-\mu \}=hP[(X-\mu )s]$. Subtracting it from step 1.4 gives $N(0,{\sigma }^2)$ for every fixed $h\ge 0$, so the sample mean is regular on every allowed path. Its influence covariance is ${\sigma }^2=V_{\mathrm{c}\mathrm{a}\mathrm{n}}$ by step 1.2; [F8] shows it attains the regular asymptotically linear lower bound. This includes ${\sigma }^2=0$, when both covariance matrices vanish.