Local asymptotic normality says that, within \(n^{-1/2}\) neighborhoods, a regular statistical experiment has the same log-likelihood geometry as observing a Gaussian shift. Contiguity identifies sequences of laws that share asymptotically negligible events, allowing stochastic expansions proved under the null to remain usable under local alternatives.
Local likelihood expansion
Definition (LAN). A sequence of experiments \[ \mathcal E_n=(\mathcal X_n,\mathcal A_n, \{P_{\theta}^{(n)}:\theta\in\Theta\}) \] is locally asymptotically normal at an interior point \(\theta_0\in\mathbb R^d\) if there are random vectors \(\Delta_n\) and a positive definite matrix \(I_{\theta_0}\) such that, for every fixed \(h\in\mathbb R^d\), \[ \log\frac{\mathrm{d}P_{\theta_0+h/\sqrt n}^{(n)}} {\mathrm{d}P_{\theta_0}^{(n)}} =h^\top\Delta_n-\frac12h^\top I_{\theta_0}h +o_{P_{\theta_0}^{(n)}}(1), \] and \[ \Delta_n\rightsquigarrow N(0,I_{\theta_0}) \quad\text{under }P_{\theta_0}^{(n)}. \]
Uniform LAN strengthens the remainder to be uniform over \(h\) in compact sets. This is needed when maximizing over a local parameter set rather than evaluating one fixed \(h\).
Quadratic-mean differentiability implies LAN
Theorem. Let \(\{P_\theta:\theta\in\Theta\}\) be an i.i.d. model differentiable in quadratic mean at \(\theta_0\), with score \(\dot\ell_{\theta_0}\) and information \[ I_{\theta_0} =P_{\theta_0} \dot\ell_{\theta_0}\dot\ell_{\theta_0}^\top. \] Then the product experiments are LAN with \[ \Delta_n=\frac1{\sqrt n}\sum_{i=1}^n \dot\ell_{\theta_0}(X_i). \]
Proof outline. Put \(h_n=h/\sqrt n\) and define the square-root likelihood increment \[ r_{n}(x) =2\left\{ \sqrt{\frac{p_{\theta_0+h_n}(x)} {p_{\theta_0}(x)}}-1 \right\}. \] Quadratic-mean differentiability gives \[ r_n=h_n^\top\dot\ell_{\theta_0} +o_{L^2(P_{\theta_0})}(n^{-1/2}), \] together with \[ P_{\theta_0}r_n =-\frac14h_n^\top I_{\theta_0}h_n+o(n^{-1}). \] Expand \[ 2\log(1+r_n/2) =r_n-\frac14r_n^2+\text{remainder} \] and sum over observations. Truncation controls the remainder, the linear terms produce \(h^\top\Delta_n\), and the quadratic terms converge to \(\frac12h^\top I_{\theta_0}h\). The multivariate central limit theorem gives the limit of \(\Delta_n\).
Example (normal location). If \(X_i\sim N(\theta,\sigma^2)\), then exactly \[ \log\frac{\mathrm{d}P_{\theta_0+h/\sqrt n}^{(n)}} {\mathrm{d}P_{\theta_0}^{(n)}} =\frac h{\sigma^2} \frac1{\sqrt n}\sum_{i=1}^n(X_i-\theta_0) -\frac{h^2}{2\sigma^2}. \] Thus \(\Delta_n=\sigma^{-2}n^{-1/2}\sum_i(X_i-\theta_0)\) and \(I=\sigma^{-2}\), with no remainder.
Contiguity
Definition. A sequence \(Q_n\) is contiguous with respect to \(P_n\), written \(Q_n\triangleleft P_n\), if \[ P_n(A_n)\to0 \quad\Longrightarrow\quad Q_n(A_n)\to0 \] for every sequence of measurable events \(A_n\). The sequences are mutually contiguous if both \(Q_n\triangleleft P_n\) and \(P_n\triangleleft Q_n\).
If \(Z_n=o_{P_n}(1)\) and \(Q_n\triangleleft P_n\), then \(Z_n=o_{Q_n}(1)\), because each event \(\{|Z_n|>\varepsilon\}\) is \(P_n\)-negligible. This transfer property is the practical reason contiguity accompanies local likelihood theory.
Le Cam's first lemma. Let \(Q_n\ll P_n\) with likelihood ratio \(L_n=\mathrm{d}Q_n/\mathrm{d}P_n\). If \[ L_n\rightsquigarrow L \quad\text{under }P_n, \qquad L>0\ \text{almost surely},\qquad \mathbb EL=1, \] then \(P_n\) and \(Q_n\) are mutually contiguous.
Proof. Since \(L_n\ge0\), \(\mathbb E_{P_n}L_n=1\), and the limiting expectation is also one, convergence in distribution implies uniform integrability. If \(P_n(A_n)\to0\), then for \(M>0\), \[ Q_n(A_n) =\mathbb E_{P_n}(L_n1_{A_n}) \le MP_n(A_n) +\mathbb E_{P_n}\{L_n1_{L_n>M}\}. \] Let \(n\to\infty\), then \(M\to\infty\), to obtain \(Q_n(A_n)\to0\). For the reverse direction, if \(Q_n(A_n)\to0\), then for \(\eta>0\), \[ P_n(A_n) \le P_n(L_n\le\eta) +\eta^{-1}Q_n(A_n). \] The first term converges to \(P(L\le\eta)\), which tends to zero as \(\eta\downarrow0\) because \(L>0\) almost surely.
Local alternatives are contiguous. Under LAN, for every fixed \(h\), \[ P_{\theta_0+h/\sqrt n}^{(n)} \quad\text{and}\quad P_{\theta_0}^{(n)} \] are mutually contiguous.
Proof. Under the null, the log-likelihood ratio converges to \[ h^\top Z-\frac12h^\top Ih, \qquad Z\sim N(0,I). \] Its exponential is strictly positive and has expectation one by the Gaussian moment-generating formula. Apply the first lemma.
Le Cam's third lemma
Third lemma. Suppose under \(P_n\), \[ \left(T_n,\log\frac{\mathrm{d}Q_n}{\mathrm{d}P_n}\right) \rightsquigarrow (T,V), \] with \(\mathbb Ee^V=1\). Then under \(Q_n\), \(T_n\) converges to the exponentially tilted law \[ \mathbb E_Qf(T)=\mathbb E\{f(T)e^V\} \] for every bounded continuous \(f\).
Proof. For bounded continuous \(f\), \[ \mathbb E_{Q_n}f(T_n) =\mathbb E_{P_n}\!\left[ f(T_n)\exp\!\left\{ \log\frac{\mathrm{d}Q_n}{\mathrm{d}P_n} \right\}\right]. \] Truncate the exponential at a fixed level, apply joint weak convergence, and then remove the truncation using uniform integrability of likelihood ratios. The limit is \(\mathbb E\{f(T)e^V\}\).
Gaussian shift under LAN. If jointly under \(P_{\theta_0}^{(n)}\), \[ \begin{pmatrix}T_n\\ \Delta_n\end{pmatrix} \rightsquigarrow N\!\left[ \begin{pmatrix}\mu\\0\end{pmatrix}, \begin{pmatrix} \Sigma&C\\ C^\top&I \end{pmatrix} \right], \] then under \(P_{\theta_0+h/\sqrt n}^{(n)}\), \[ T_n\rightsquigarrow N(\mu+Ch,\Sigma). \]
Proof. Exponential tilting of a joint Gaussian by \[ e^{h^\top\Delta-\frac12h^\top Ih} \] shifts its mean by its covariance with \(h^\top\Delta\). The \(T\)-component therefore shifts by \(Ch\), while covariance remains unchanged.
Local tests and Gaussian limit experiment
The LAN limit is equivalent to observing \[ Y=I_{\theta_0}h+Z,\qquad Z\sim N(0,I_{\theta_0}), \] or, after whitening, \(I_{\theta_0}^{1/2}h+N(0,I_d)\). The likelihood-ratio test, efficient estimator, and local minimax risk of the original experiment converge to their Gaussian-shift counterparts under suitable convergence-of-experiments conditions. Local alternatives at scales larger than \(n^{-1/2}\) are often consistently distinguishable; smaller scales are asymptotically invisible in regular directions.
When LAN fails
Moving supports, change-points, mixture singularities, weak identification, cusp models, and boundaries can produce different local rates and nonquadratic likelihood processes. In such problems, forcing a Fisher-information expansion suppresses the very feature that determines the limit experiment.