Le Cam's lemmas are change-of-measure theorems for triangular arrays. The first lemma recognizes contiguity from a likelihood-ratio limit. The third lemma tilts a joint null limit to obtain the law under contiguous alternatives. A Gaussian specialization turns exponential tilting into a mean shift.
Le Cam's first lemma
Theorem. Let \(Q_n\ll P_n\) and \(L_n=\mathrm{d}Q_n/\mathrm{d}P_n\). Suppose \[ L_n\rightsquigarrow L \quad\text{under }P_n. \] Then:
- \(Q_n\triangleleft P_n\) if and only if \(EL=1\);
- if \(EL=1\) and \(P(L>0)=1\), then \(P_n\) and \(Q_n\) are mutually contiguous.
Proof. Since \(E_{P_n}L_n=1\), weak convergence and nonnegativity give \(EL\leq1\). If \(EL=1\), convergence in distribution plus convergence of first moments implies uniform integrability of \(L_n\), hence \(Q_n\triangleleft P_n\) by the preceding chapter. Conversely, contiguity is equivalent to uniform integrability, and uniform integrability plus weak convergence gives \(EL=\lim E_{P_n}L_n=1\).
For the second assertion, the law of \(L\) under the limiting \(Q\)-measure is tilted by \(L\). If \(L>0\) almost surely, the reciprocal \(1/L\) is finite under that tilted law and has mean one: \[ E_Q(1/L)=E_P\{L(1/L)\}=1. \] Applying the first assertion in the reverse direction yields \(P_n\triangleleft Q_n\). \(\square\)
Example (lost mass). Let \(P_n(A_n)=1/n\) and set \(L_n=n1_{A_n}\). Then \(E_{P_n}L_n=1\), but \(L_n\to0\) in \(P_n\)-probability, so \(EL=0\). The measure \(Q_n\) puts all mass on \(A_n\), and is not contiguous with respect to \(P_n\). This is exactly probability mass escaping through rare, enormous likelihood ratios.
Le Cam's third lemma
Theorem. Suppose \(Q_n\ll P_n\) and, under \(P_n\), \[ (T_n,L_n)\rightsquigarrow(T,L), \qquad L_n=\frac{\mathrm{d}Q_n}{\mathrm{d}P_n}, \qquad EL=1. \] Then under \(Q_n\), \[ T_n\rightsquigarrow T^{(Q)}, \] where the limiting law is defined by \[ E f(T^{(Q)})=E\{Lf(T)\} \] for every bounded continuous \(f\).
Proof. For bounded continuous \(f\), \[ E_{Q_n}f(T_n)=E_{P_n}\{L_nf(T_n)\}. \] The product map is unbounded in \(L_n\), so joint weak convergence alone is insufficient. The condition \(EL=1\) and the first lemma imply uniform integrability of \(L_n\). Truncate \(L_n\) at \(M\), use bounded continuous approximation and joint convergence, then let \(M\to\infty\) uniformly. This gives \[ E_{Q_n}f(T_n)\to E\{Lf(T)\}. \] The right side is a probability expectation because \(L\geq0\) and \(EL=1\). \(\square\)
Gaussian exponential tilting
Theorem (Gaussian form of the third lemma). Suppose under \(P_n\), \[ \begin{pmatrix}T_n\\ \log L_n\end{pmatrix} \rightsquigarrow N\!\left[ \begin{pmatrix}\mu\\-\frac12\tau^2\end{pmatrix}, \begin{pmatrix} \Sigma&c\\ c^\top&\tau^2 \end{pmatrix} \right]. \] Then under \(Q_n\), \[ T_n\rightsquigarrow N(\mu+c,\Sigma). \]
Proof. The limiting likelihood ratio is \(L=e^V\), where \(V\sim N(-\tau^2/2,\tau^2)\), so \(EL=1\). For \(u\in\mathbb R^d\), \[ E\{e^Ve^{iu^\top T}\} = \exp\left\{ iu^\top(\mu+c)-\frac12u^\top\Sigma u \right\}. \] This follows from the joint Gaussian moment-generating function evaluated at \((iu,1)\). The expression is the characteristic function of \(N(\mu+c,\Sigma)\); apply the third lemma. \(\square\)
LAN alternatives
Corollary. Suppose a model is LAN: \[ \log\frac{\mathrm{d}P_{n,h}}{\mathrm{d}P_{n,0}} = h^\top\Delta_n-\frac12h^\top Ih+o_{P_{n,0}}(1), \qquad \Delta_n\rightsquigarrow N(0,I). \] Then \(P_{n,h}\triangleleft\triangleright P_{n,0}\), and under \(P_{n,h}\), \[ \Delta_n\rightsquigarrow N(Ih,I). \]
Proof. Under the null, the likelihood ratio converges to \[ L_h=\exp\{h^\top Z-\tfrac12h^\top Ih\}, \qquad Z\sim N(0,I). \] It is positive and has mean one, so the first lemma gives mutual contiguity. The covariance between \(Z\) and \(h^\top Z\) is \(Ih\); the Gaussian third lemma shifts the mean accordingly. \(\square\)
Local power
Example. If a scalar statistic satisfies jointly \[ (T_n,\Delta_n)\rightsquig N\!\left( 0, \begin{pmatrix}\sigma^2&c^\top\\c&I\end{pmatrix} \right) \] under \(P_{n,0}\), then under \(P_{n,h}\), \[ T_n\rightsquig N(c^\top h,\sigma^2). \] A level-\(\alpha\) test rejecting for \(T_n>\sigma z_{1-\alpha}\) has local limiting power \[ 1-\Phi\!\left( z_{1-\alpha}-\frac{c^\top h}{\sigma} \right). \]
The covariance \(c\) measures how strongly the statistic aligns with the score direction. Efficient tests maximize this alignment subject to their null variance.