LAN is centered at one fixed parameter. A procedure intended to work over a whole parameter space must first discover the relevant local chart and then apply the local Gaussian rule there. Global asymptotic normality is obtained by localization, uniform local approximation, and patching.
Uniform local asymptotic normality
Definition. Let \(K\subset\Theta^\circ\) be compact. A model is uniformly LAN on \(K\) if, uniformly for \(\theta\in K\) and \(h\) in compact sets, \[ \log\frac{\mathrm{d}P_{\theta+h/\sqrt n}^{\,n}} {\mathrm{d}P_\theta^{\,n}} = h^\top\Delta_{n,\theta} -\frac12h^\top I(\theta)h +o_{P_\theta}(1), \] and \[ \Delta_{n,\theta}\rightsquigarrow N(0,I(\theta)) \] uniformly in an appropriate bounded-Lipschitz sense.
Pointwise LAN at every \(\theta\) does not imply uniform LAN. Uniform control of score moments, information continuity, and likelihood remainders is needed to use a random center.
Preliminary localization
Definition. A preliminary estimator \(\tilde\theta_n\) localizes uniformly on \(K\) at rate \(a_n^{-1}\) if \[ \lim_{M\to\infty}\limsup_n \sup_{\theta\in K} P_\theta\{a_n\|\tilde\theta_n-\theta\|>M\}=0. \]
For a one-step efficient correction, the preliminary rate need only satisfy \[ \|\tilde\theta_n-\theta\|=o_p(n^{-1/4}) \] under a smooth score expansion; it need not already be root-\(n\) efficient.
Sample splitting and random centers
A clean construction uses a small first subsample to obtain \(\tilde\theta_n\) and the remaining observations for the local update. Conditional on the first subsample, the center is fixed relative to the second, so uniform LAN can be applied without a same-sample random-index problem. If the first subsample size tends to infinity but is \(o(n)\), it can localize consistently while sacrificing asymptotically negligible information.
Theorem (one-step globalization). Let \(\tilde\theta_n\) satisfy \(\|\tilde\theta_n-\theta\|=o_{P_\theta}(n^{-1/4})\) uniformly on compact \(K\). Suppose the score \(S_n(\theta)\) and observed information \(J_n(\theta)\) satisfy uniformly near \(K\), \[ n^{-1}J_n(\theta)\to_p I(\theta), \] \[ n^{-1/2}\{S_n(\tilde\theta_n)-S_n(\theta)\} = -I(\theta)\sqrt n(\tilde\theta_n-\theta) +o_p(1). \] Then \[ \hat\theta_n = \tilde\theta_n+ J_n(\tilde\theta_n)^{-1}S_n(\tilde\theta_n) \] satisfies \[ \sqrt n(\hat\theta_n-\theta) = I(\theta)^{-1}\frac{S_n(\theta)}{\sqrt n} +o_{P_\theta}(1) \] uniformly on \(K\).
Proof. Write \(J_n(\tilde\theta_n)^{-1}=n^{-1}I(\theta)^{-1}+o_p(n^{-1})\) uniformly. The score expansion gives \[ S_n(\tilde\theta_n) = S_n(\theta)-nI(\theta)(\tilde\theta_n-\theta)+o_p(\sqrt n). \] Substitution into the update cancels the preliminary error: \[ \hat\theta_n-\theta = \tilde\theta_n-\theta \frac1nI(\theta)^{-1}S_n(\theta) -(\tilde\theta_n-\theta) +o_p(n^{-1/2}). \] Multiply by \(\sqrt n\). \(\square\)
Patching local procedures
More abstractly, cover a compact parameter set by neighborhoods \(U_{n,j}\) on which local experiments are close to Gaussian shifts. A preliminary statistic selects an index \(j\), and a local Markov kernel or local decision rule is applied. If the selection is correct with high probability and approximation errors are uniform, the patched global procedure inherits the local risks.
Proposition. If local simulation kernels have errors at most \(\varepsilon_n\) uniformly on each selected chart and chart-selection failure has probability at most \(\eta_n\), then every bounded-loss risk transferred through the patched kernel differs by at most \[ \varepsilon_n+\eta_n. \]
The bound follows by splitting according to correct localization and applying total-variation risk transfer on that event.
Locally asymptotically mixed normal experiments
Definition. A model is LAMN if \[ \log\frac{\mathrm{d}P_{n,h}}{\mathrm{d}P_{n,0}} = h^\top\Delta_n-\frac12h^\top J_nh+o_p(1), \] where \[ (\Delta_n,J_n)\rightsquigarrow(J^{1/2}Z,J), \qquad Z\sim N(0,I_p),\quad Z\perp J, \] and \(J\) is random positive definite.
The limit experiment first reveals random information \(J\), then observes a Gaussian shift with conditional information \(J\). Stable convergence, rather than ordinary weak convergence, records the joint relation with ancillary randomness. Mixed normal limits arise in diffusion models, stochastic volatility, and heteroskedastic experiments.
Global limitations
Boundary points may have different rates or non-Gaussian limits; information may degenerate; parameter spaces may require incompatible charts; and preliminary estimators may not localize uniformly. Global claims must therefore state the compact sets or strata on which uniform approximation holds. A collection of pointwise normal limits is not itself a global asymptotic experiment.