45.04.07 · mathematical-statistics / 04-asymptotic-statistics

Contiguity and Le Cam's Three Lemmas

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Anchor (Master): van der Vaart 1998 Asymptotic Statistics (Cambridge) Ch. 6 (contiguity §6.1, the first, second, and third lemmas §6.2-6.4, applications to local power §6.5) and Ch. 7 (the LAN log-likelihood expansion that supplies the asymptotically normal log-ratio); Le Cam & Yang 2000 Asymptotics in Statistics 2e (Springer) Ch. 3 (contiguity, the relation to Hellinger affinities and the Kakutani dichotomy) and Ch. 6 (the LAN consequences); Le Cam 1960 Locally asymptotically normal families (Univ. Calif. Publ. Statist. 3) §§3-5 (the original first and third lemmas); Hájek, Šidák & Sen 1999 Theory of Rank Tests 2e (Academic Press) Ch. 7 (contiguity machinery for local power of rank tests)

Intuition Beginner

Two ways of assigning probabilities to the same data can be close or far apart. If they are close, an event that one of them calls nearly impossible the other also calls nearly impossible; if they are far apart, each can call certain what the other calls impossible. Contiguity is the large-sample version of "close." You have a sequence of null laws, one for each sample size, and a matching sequence of alternative laws drifting toward the null. Contiguity says the alternatives never run away: whatever the null deems negligible in the long run, the alternatives also deem negligible.

Why bother? Because it lets you reuse work. Suppose you have computed how some test statistic behaves under the null — its bell curve, its centre, its spread. You would like the same description under a nearby alternative, where the truth might actually lie. Without contiguity you would redo the whole calculation. With contiguity, and one extra ingredient — how the statistic moves together with the likelihood ratio — you simply slide the null bell curve over by a fixed amount and you are done.

The likelihood ratio is the bookkeeper. It records, sample by sample, how much more plausible the alternative makes the data than the null does. Contiguity is exactly the condition that this bookkeeper does not blow up or vanish in a way that severs the two worlds. When it stays well behaved, the two worlds are different views of one scene, and you can pass freely between them.

The takeaway: contiguity is asymptotic absolute continuity; once you have it, the limit behaviour of any statistic under a nearby alternative is the null behaviour shifted by the statistic's covariance with the log-likelihood ratio — no recomputation needed.

Visual Beginner

Picture two bell curves for the same test statistic. The first is its distribution under the null: a bell centred at zero. The second is its distribution under a nearby alternative. The lesson of contiguity is that the second bell is the first one slid sideways by a fixed amount, with the same width. You do not reshape the curve; you only translate it. The size of the slide is one number — how strongly the statistic and the likelihood-ratio bookkeeper move together.

question	under the null	under the contiguous alternative
centre of the statistic	known, say zero	zero plus the shift
width of the statistic	known	the same width
the shift	—	covariance of statistic with the log-ratio
how you get it	direct calculation	translate the null answer

The takeaway: contiguity turns "find the alternative distribution" into "translate the null distribution," and the only new number you need is one covariance.

Worked example Beginner

We watch the shift appear for a normal mean. Each observation is normal with variance one and unknown mean. The null says the mean is zero; the alternative puts the mean at a small value $h / n$ , drifting toward zero as the sample grows. The statistic of interest is the standardised sample average $T_{n} = n \overset{ˉ}{X}_{n}$ .

Step 1. The null behaviour. Under the null (mean zero), $T_{n} = n \overset{ˉ}{X}_{n}$ has mean zero and variance one for every $n$ — a standard bell curve centred at zero.

Step 2. The bookkeeper. The log-likelihood ratio for mean $h / n$ against mean zero works out, for this model, to $Λ_{n} = h T_{n} - \frac{1}{2} h^{2}$ . So the statistic and the log-ratio are linked: $Λ_{n}$ is just $h$ times $T_{n}$ , minus a constant.

Step 3. The shift. The shift is how $T_{n}$ and $Λ_{n}$ move together. Since $Λ_{n} = h T_{n} - \frac{1}{2} h^{2}$ and $T_{n}$ has variance one, the covariance of $T_{n}$ with $Λ_{n}$ is $h \times 1 = h$ .

Step 4. Read off the alternative. By the shift rule, under the alternative $T_{n}$ has the same bell shape, centred now at $h$ instead of zero: variance still one, mean now $h$ . Take $h = 2$ : under the null $T_{n}$ centres at $0$ ; under the alternative it centres at $2$ , same spread.

What this tells us: we never recomputed a distribution under the alternative. We took the null bell curve, found one covariance ( $h$ ), and slid the curve over by that amount. That slide is the whole content of the shift rule.

Check your understanding Beginner

Exercise (easy, multiple choice).

Contiguity of a sequence of alternative laws with respect to a sequence of null laws says, in plain terms:

A. The alternative and null laws are eventually equal B. Whatever the null laws call negligible in the limit, the alternative laws also call negligible C. The alternative laws have larger variance than the null laws D. The likelihood ratio converges to zero

Hint

Contiguity is the large-sample version of absolute continuity: a one-directional "no escaping mass" condition.

Answer

B. Contiguity means events whose null probability goes to zero also have alternative probability going to zero — the alternatives never put their mass where the null has none. Feedback-correct: this is exactly asymptotic absolute continuity, the condition that lets you transfer limit statements across the two laws. Feedback-wrong: A is far too strong (the laws differ, only their negligible sets align); C is about spread, not contiguity; D would actually break the link, since a vanishing likelihood ratio signals singularity.

Formal definition Intermediate+

Let $(X_{n}, A_{n})$ be measurable spaces and let $P_{n}, Q_{n}$ be probability measures on $(X_{n}, A_{n})$ . Convergence in distribution $d$ , convergence in probability $P$ , and the $o_{P} / O_{P}$ calculus are as in 45.04.01; the LAN expansion and the central sequence $Δ_{n, θ_{0}}$ are as in 45.04.06.

Definition (contiguity). The sequence $(Q_{n})$ is contiguous with respect to $(P_{n})$ , written $Q_{n} ◃ P_{n}$ , if for every sequence of events $A_{n} \in A_{n}$ ,

P_{n} (A_{n}) \to 0 ⟹ Q_{n} (A_{n}) \to 0.

The sequences are mutually contiguous, written $Q_{n} ◃ ▹ P_{n}$ , if $Q_{n} ◃ P_{n}$ and $P_{n} ◃ Q_{n}$ . Contiguity is the asymptotic analogue of absolute continuity: $Q ≪ P$ means $P (A) = 0 \Rightarrow Q (A) = 0$ , and contiguity replaces the single pair by sequences and " $= 0$ " by " $\to 0$ ." A fixed pair with $Q ≪ P$ gives the constant sequences $Q_{n} \equiv Q$ , $P_{n} \equiv P$ a contiguous sequence; the genuinely new phenomenon is that $Q_{n}$ and $P_{n}$ may be mutually singular for every fixed $n$ yet still be contiguous as sequences.

Definition (likelihood ratio and its escaping mass). Assume $Q_{n} ≪ P_{n}$ (the dominated case; the general case adds the singular part below). Write $W_{n} = d Q_{n} / d P_{n}$ for the likelihood ratio, so $E_{P_{n}} W_{n} = 1$ . A sequence of laws loses mass to $\infty$ under $P_{n}$ if there exist $M_{k} \to \infty$ with $lim inf_{n} Q_{n} (W_{n} > M_{k}) > 0$ ; equivalently the $P_{n}$ -laws of $W_{n}$ are not uniformly tight after the change of measure. The relation to the Hellinger affinity $ρ_{n} = \int d P_{n} d Q_{n}$ of 37.04.04 is direct: $ρ_{n} = E_{P_{n}} W_{n}$ , and $ρ_{n}$ bounded away from $0$ is the affinity form of the no-escaping-mass condition, the asymptotic shadow of the Kakutani dichotomy.

Definition (the limit log-likelihood ratio under LAN). When $(P_{n}, Q_{n}) = (P_{θ_{0}}^{\otimes n}, P_{θ_{0} + h / n}^{\otimes n})$ comes from a LAN family at $θ_{0}$ 45.04.06, the log-likelihood ratio is

Λ_{n} = lo g \frac{d Q _{n}}{d P _{n}} = ⟨ h, Δ_{n, θ_{0}} ⟩ - \frac{1}{2} ⟨ h, I (θ_{0}) h ⟩ + o_{P_{n}} (1),

and under $P_{n}$ , $Λ_{n} d N (- \frac{1}{2} σ^{2}, σ^{2})$ with $σ^{2} = ⟨ h, I (θ_{0}) h ⟩$ . The mean of the limit equals $- \frac{1}{2}$ its variance: the signature of a contiguous log-ratio. This $N (- \frac{1}{2} σ^{2}, σ^{2})$ law is the canonical limit that drives the lemmas below.

Counterexamples to common slips Intermediate+

Contiguity is directional. $Q_{n} ◃ P_{n}$ does not entail $P_{n} ◃ Q_{n}$ . If $Q_{n}$ concentrates on a set the null gives small but not vanishing probability while the null spreads onto a set the alternative ignores, one direction can hold and the other fail. Mutual contiguity is the symmetric, two-sided statement, and only it licenses transferring limits in both directions.
Fixed- $n$ singularity is compatible with contiguity. For the Gaussian shift $P_{n} = N (0, I_{n})^{\otimes}$ versus $Q_{n} = N (μ_{n}, I_{n})$ with $\sum_{k} μ_{n, k}^{2} \to c < \infty$ , the laws can be mutually singular at each $n$ yet contiguous as sequences. Contiguity is a statement about sequences, not about any single pair; reading it as fixed- $n$ absolute continuity is the standard error.
A degenerate limit log-ratio breaks contiguity. If $Λ_{n} d Λ$ with $E [e^{Λ}] < 1$ , then mass has escaped to $- \infty$ (equivalently the reciprocal ratio escapes to $+ \infty$ ), and $Q_{n}$ is not contiguous to $P_{n}$ . The criterion is exactly $E [e^{Λ}] = 1$ ; an inequality, not just convergence, is required.
The shift in the third lemma is a covariance, not a mean. The amount by which a statistic's limit centre moves under the alternative is $Cov (T, Λ)$ in the joint null limit, not the alternative mean of anything computed separately. A statistic asymptotically independent of the log-ratio ( $Cov (T, Λ) = 0$ ) has the same limit under null and alternative despite the laws differing.

Key theorem with proof Intermediate+

The signature result is Le Cam's first lemma: contiguity is completely characterised by the limit law of the likelihood ratio, and in particular by whether that limit conserves total mass. The proof reduces contiguity to uniform integrability of the likelihood ratios via Prohorov's tightness theorem, then identifies the mass-conservation condition $E [e^{Λ}] = 1$ as the operative criterion. The result is the asymptotic counterpart of the Kakutani affinity dichotomy 37.04.04.

Theorem (Le Cam's first lemma; van der Vaart Ch. 6). Let $P_{n}, Q_{n}$ be probability measures with $Q_{n} ≪ P_{n}$ and likelihood ratio $W_{n} = d Q_{n} / d P_{n}$ . The following are equivalent.

(a) $Q_{n} ◃ P_{n}$ .

(b) If $W_{n} d W$ along a subsequence under $P_{n}$ , then $E [W] = 1$ .

(c) If $1/ W_{n} = d P_{n} / d Q_{n} d V$ along a subsequence under $Q_{n}$ , then $V$ places no mass at $0$ , i.e. $P (V = 0) = 0$ .

In particular, when the log-ratio satisfies $Λ_{n} = lo g W_{n} d N (- \frac{1}{2} σ^{2}, σ^{2})$ under $P_{n}$ , contiguity holds, because $E [e^{Λ}] = exp (- \frac{1}{2} σ^{2} + \frac{1}{2} σ^{2}) = 1$ .

Proof. Write $L_{P_{n}} (W_{n})$ for the law of $W_{n}$ under $P_{n}$ .

(a) $\Rightarrow$ (b). The family ${L_{P_{n}} (W_{n})}$ is tight: if not, there are $M_{k} \to \infty$ and a subsequence with $P_{n} (W_{n} > M_{k}) \geq ε > 0$ infinitely often — but $P_{n} (W_{n} > M) \leq E_{P_{n}} W_{n} / M = 1/ M \to 0$ uniformly, so the laws are tight, and any subsequential weak limit $W$ exists. Suppose along a subsequence $W_{n} d W$ under $P_{n}$ with $E [W] < 1$ (Fatou gives $\leq 1$ always). Then mass $1 - E [W] > 0$ escapes upward: there are $M_{k} \to \infty$ with $lim_{n} E_{P_{n}} [W_{n} 1 {W_{n} > M_{k}}] \geq 1 - E [W] - o (1) > 0$ . Setting $A_{n} = {W_{n} > M_{k (n)}}$ for a slowly increasing $M_{k (n)}$ , one has $P_{n} (A_{n}) \leq 1/ M_{k (n)} \to 0$ while $Q_{n} (A_{n}) = E_{P_{n}} [W_{n} 1_{A_{n}}] \neq \to 0$ , contradicting $Q_{n} ◃ P_{n}$ . Hence $E [W] = 1$ .

(b) $\Rightarrow$ (a). Suppose $P_{n} (A_{n}) \to 0$ but, along a subsequence, $Q_{n} (A_{n}) \geq ε > 0$ . By tightness pass to a further subsequence with $W_{n} d W$ under $P_{n}$ and $E [W] = 1$ . Since $E_{P_{n}} W_{n} = 1 = E [W]$ and $W_{n} d W$ , the family ${W_{n}}$ is uniformly integrable under $P_{n}$ (convergence of means together with weak convergence of non-negative variables forces UI). Then $Q_{n} (A_{n}) = E_{P_{n}} [W_{n} 1_{A_{n}}] \to 0$ because $P_{n} (A_{n}) \to 0$ and ${W_{n}}$ is UI, contradicting $Q_{n} (A_{n}) \geq ε$ . Hence $Q_{n} ◃ P_{n}$ .

(b) $\Leftrightarrow$ (c). Under $Q_{n}$ the variable $1/ W_{n}$ is the likelihood ratio $d P_{n} / d Q_{n}$ ; mass conservation upstairs ( $E [W] = 1$ , no escape of $W_{n}$ to $+ \infty$ under $P_{n}$ ) is equivalent to no atom of the reciprocal at $0$ under $Q_{n}$ , since ${W_{n} > M}$ corresponds to ${1/ W_{n} < 1/ M}$ and the escaping $P_{n}$ -mass of $W_{n}$ is the $Q_{n}$ -mass piling up near $1/ W_{n} = 0$ . The change-of-measure bookkeeping $Q_{n} (W_{n} > M) = E_{P_{n}} [W_{n} 1 {W_{n} > M}]$ makes the two escape conditions identical.

The Gaussian case. If $Λ_{n} d Λ \sim N (- \frac{1}{2} σ^{2}, σ^{2})$ then $W_{n} = e^{Λ_{n}} d e^{Λ}$ by the continuous-mapping theorem, and for a normal $Λ$ , $E [e^{Λ}] = exp (E Λ + \frac{1}{2} Var Λ) = exp (- \frac{1}{2} σ^{2} + \frac{1}{2} σ^{2}) = 1$ . By (b) contiguity holds. $□$

Bridge. The first lemma builds toward the third lemma's transfer of limit laws across measures, and the same likelihood-ratio limit appears again in the local-power computations of 45.04.09, where the non-central chi-squared arises from exactly this contiguous change of measure. The foundational reason contiguity reduces to mass conservation is that the likelihood ratio is a non-negative mean-one variable, so its only way to break the link between $P_{n}$ and $Q_{n}$ is to leak mass to $0$ or $\infty$ ; ruling that out is precisely the uniform-integrability statement that Prohorov tightness and convergence of means supply. This is exactly the asymptotic incarnation of the Kakutani dichotomy 37.04.04: there the Hellinger product $\prod a_{k}$ stays positive or collapses, here the Hellinger affinity $ρ_{n} = E_{P_{n}} W_{n}$ stays bounded away from $0$ or not, and the $\cdot$ that linearised the product in Kakutani is the same root-likelihood that linearises the log-ratio under LAN 45.04.06. Putting these together, contiguity generalises absolute continuity from a single pair to a drifting sequence, and the bridge is that every limit statement proved under the null transports to the contiguous alternative once one knows the joint limit of the statistic with the log-ratio — which is what the third lemma packages.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

(Second lemma.) Using the LAN expansion $Λ_{n} = ⟨ h, Δ_{n} ⟩ - \frac{1}{2} h^{T} I h + o_{P} (1)$ with $Δ_{n} d N (0, I)$ under the null, derive that $Λ_{n} d N (- \frac{1}{2} σ^{2}, σ^{2})$ with $σ^{2} = h^{T} I h$ , and conclude mutual contiguity.

Hint

$⟨ h, Δ_{n} ⟩$ is a linear image of an asymptotically normal vector; compute its limiting variance, then add the deterministic shift.

Answer

Since $Δ_{n} d N (0, I)$ , the linear functional $⟨ h, Δ_{n} ⟩ d N (0, h^{T} I h) = N (0, σ^{2})$ by the continuous-mapping theorem. The deterministic term $- \frac{1}{2} h^{T} I h = - \frac{1}{2} σ^{2}$ and the $o_{P} (1)$ remainder give, by Slutsky, $Λ_{n} d N (- \frac{1}{2} σ^{2}, σ^{2})$ . The mean equals $- \frac{1}{2}$ the variance, so $E [e^{Λ}] = 1$ and the first lemma gives $Q_{n} ◃ P_{n}$ . For the reverse direction, under $Q_{n}$ the log-ratio $- Λ_{n}$ (the reciprocal) has the symmetric LAN structure with limit $N (- \frac{1}{2} σ^{2}, σ^{2})$ as well (Le Cam's third lemma applied to $Λ_{n}$ itself, or the change-of-measure computation), so $P_{n} ◃ Q_{n}$ and the sequences are mutually contiguous. Rubric: full credit for the limit law, the $E [e^{Λ}] = 1$ check, and the two-sided conclusion. This is Le Cam's second lemma — the LAN log-ratio is asymptotically normal — feeding the first.

Exercise 4 (medium, symbolic).

(Third lemma.) Suppose under $P_{n}$ , $(T_n, \Lambda_n) \xrightarrow{d} N\big((0, -\tfrac12\sigma^2), \begin{psmallmatrix}\tau^2 & c\\ c & \sigma^2\end{psmallmatrix}\big)$ with $Λ_{n}$ the LAN log-ratio. Show that under the contiguous alternative $Q_{n}$ , $T_{n} d N (c, τ^{2})$ .

Hint

For bounded continuous $f$ , $E_{Q_{n}} f (T_{n}) = E_{P_{n}} [f (T_{n}) e^{Λ_{n}}]$ ; tilting a bivariate Gaussian by $e^{Λ}$ shifts the mean of $T$ by $Cov (T, Λ)$ .

Answer

Contiguity (first/second lemmas) gives uniform integrability of ${e^{Λ_{n}}}$ under $P_{n}$ , so for bounded continuous $f$ , $E_{Q_{n}} f (T_{n}) = E_{P_{n}} [f (T_{n}) e^{Λ_{n}}] \to E [f (T) e^{Λ}]$ by the joint convergence. For jointly Gaussian $(T, Λ)$ with $E [e^{Λ}] = 1$ , the tilted law $f \mapsto E [f (T) e^{Λ}]$ is Gaussian with unchanged covariance $τ^{2}$ and mean shifted by $Cov (T, Λ) = c$ : completing the square in the bivariate density against $e^{λ}$ moves $E [T]$ from $0$ to $0 + c$ . Hence under $Q_{n}$ , $T_{n} d N (c, τ^{2})$ . Rubric: full credit for the change-of-measure identity, the Gaussian tilt, and the shift by $c$ . This is Le Cam's third lemma, the engine of all local-power computations.

Exercise 5 (medium, symbolic).

Show that contiguity is not symmetric in general by exhibiting sequences with $Q_{n} ◃ P_{n}$ but $P_{n} \neq ◃ Q_{n}$ . Take $P_{n} = N (0, 1)$ and $Q_{n} = (1 - 1/ n) N (0, 1) + (1/ n) N (a_{n}, 1)$ with $a_{n} \to \infty$ chosen suitably.

Hint

Make $Q_{n}$ a small contamination of $P_{n}$ that escapes to infinity; check the two directions of the negligible-set condition.

Answer

Let $P_{n} = N (0, 1)$ and $Q_{n} = (1 - 1/ n) N (0, 1) + (1/ n) N (a_{n}, 1)$ with $a_{n} = n$ . For $A_{n}$ with $P_{n} (A_{n}) \to 0$ : $Q_{n} (A_{n}) = (1 - 1/ n) P_{n} (A_{n}) + (1/ n) N (a_{n}, 1) (A_{n}) \leq P_{n} (A_{n}) + 1/ n \to 0$ , so $Q_{n} ◃ P_{n}$ (the contaminating mass $1/ n$ vanishes). For the reverse, take $A_{n} = {x > a_{n} /2}$ . Then $P_{n} (A_{n}) = \overset{ˉ}{Φ} (a_{n} /2) \to 0$ , but $Q_{n} (A_{n}) \geq (1/ n) N (a_{n}, 1) (x > a_{n} /2) = (1/ n) Φ (a_{n} /2) \to 0$ as well here — so refine: take $A_{n} = {x > a_{n} - 1}$ and scale the contamination weight to $1/ n$ instead, with $a_{n}$ growing so that $n \overset{ˉ}{Φ} (a_{n}) \to 0$ while $(1/ n) Φ (\cdot)$ on the bump stays $\geq ε$ ; then $P_{n} (A_{n}) \to 0$ but $Q_{n} (A_{n}) \neq \to 0$ , so $P_{n} \neq ◃ Q_{n}$ . The asymmetry is that $Q_{n}$ keeps a non-vanishing-relative bump in a region the null calls negligible. Rubric: full credit for one valid construction making one direction hold and the other fail, with both negligible-set checks. The point is that mutual contiguity is strictly stronger than one-sided.

Exercise 6 (medium, symbolic).

Relate contiguity to the Hellinger affinity. For product measures $P_{n} = ⨂_{k \leq n} p_{k}$ and $Q_{n} = ⨂_{k \leq n} q_{k, n}$ show that if $lim inf_{n} \prod_{k \leq n} H (p_{k}, q_{k, n}) > 0$ then $Q_{n} ◃ ▹ P_{n}$ , connecting to the Kakutani dichotomy of 37.04.04.

Hint

The affinity $ρ_{n} = E_{P_{n}} W_{n} = \prod_{k \leq n} H (p_{k}, q_{k, n})$ ; a Hellinger lower bound controls the escaping mass of $W_{n}$ .

Answer

By independence the Hellinger affinity factorises: $ρ_{n} = \int d P_{n} d Q_{n} = \prod_{k \leq n} \int p_{k} q_{k, n} d μ = \prod_{k \leq n} H (p_{k}, q_{k, n})$ , and $ρ_{n} = E_{P_{n}} W_{n}$ with $W_{n} = d Q_{n} / d P_{n}$ . The Hellinger distance satisfies $h^{2} (P_{n}, Q_{n}) = 1 - ρ_{n}$ , so $ρ_{n} \geq δ > 0$ bounds the distance away from its maximum. A standard inequality gives, for any event $A_{n}$ , $∣ Q_{n} (A_{n}) - P_{n} (A_{n}) ∣ \leq 2 h (P_{n}, Q_{n}) \leq 2 (1 - δ)$ uniformly, and more precisely $ρ_{n}$ bounded away from $0$ prevents $W_{n}$ from escaping to $0$ or $\infty$ (the limit law of $W_{n}$ conserves mean one), so by the first lemma both directions of contiguity hold. Rubric: full credit for the affinity factorisation, the identification $ρ_{n} = E_{P_{n}} W_{n}$ , and the no-escaping-mass conclusion. This is the affinity form of Kakutani's dichotomy 37.04.04 made into a contiguity statement: a positive limiting Hellinger product is asymptotic mutual absolute continuity.

Exercise 7 (hard, symbolic).

(Local power of the score test.) In a LAN family at $θ_{0}$ , the score test rejects when $T_{n} = (h_{0}^{T} I h_{0})^{- 1/2} ⟨ h_{0}, Δ_{n, θ_{0}} ⟩$ exceeds $z_{1 - α}$ , for a fixed direction $h_{0}$ . Under the null $T_{n} d N (0, 1)$ . Use the third lemma to compute the limiting power under the contiguous alternative $θ_{0} + h_{0} / n$ , and show it is $1 - Φ (z_{1 - α} - δ)$ with $δ = (h_{0}^{T} I h_{0})^{1/2}$ .

Hint

Compute the joint null limit of $(T_{n}, Λ_{n})$ with $Λ_{n} = ⟨ h_{0}, Δ_{n} ⟩ - \frac{1}{2} h_{0}^{T} I h_{0}$ , find $Cov (T, Λ)$ , and apply the third-lemma shift.

Answer

Write $σ^{2} = h_{0}^{T} I h_{0}$ and $T_{n} = σ^{- 1} ⟨ h_{0}, Δ_{n} ⟩$ . Under $P_{n}$ , $(T_{n}, Λ_{n}) d (T, Λ)$ jointly Gaussian with $T \sim N (0, 1)$ , $Λ \sim N (- \frac{1}{2} σ^{2}, σ^{2})$ , and $Cov (T, Λ) = σ^{- 1} Cov (⟨ h_{0}, Δ ⟩, ⟨ h_{0}, Δ ⟩ - \frac{1}{2} σ^{2}) = σ^{- 1} \cdot σ^{2} = σ$ . By the third lemma, under the contiguous alternative $T_{n} d N (σ, 1)$ , i.e. the standard normal shifted to mean $δ = σ = (h_{0}^{T} I h_{0})^{1/2}$ . The limiting power is therefore

n lim Q_{n} (T_{n} > z_{1 - α}) = P (N (δ, 1) > z_{1 - α}) = 1 - Φ (z_{1 - α} - δ) .

Rubric: full credit for the joint Gaussian limit, the covariance $Cov (T, Λ) = σ$ , the third-lemma shift to $N (δ, 1)$ , and the power formula. The crucial saving is that the alternative distribution of $T_{n}$ was obtained by translating its null $N (0, 1)$ law, never by recomputing the score statistic's sampling distribution under $θ_{0} + h_{0} / n$ .

Exercise 8 (hard, symbolic).

(Noncentral chi-squared local power, multidimensional.) Let the score-test statistic be the quadratic form $S_{n} = Δ_{n, θ_{0}}^{T} I (θ_{0})^{- 1} Δ_{n, θ_{0}}$ , with $S_{n} d χ_{k}^{2}$ under the null. Using the third lemma, show that under the contiguous alternative $θ_{0} + h / n$ , $S_{n} d χ_{k}^{2} (λ)$ , a noncentral chi-squared with noncentrality $λ = h^{T} I (θ_{0}) h$ .

Hint

Apply the third lemma componentwise to $Δ_{n}$ : under the alternative $Δ_{n} d N (I h, I)$ ; then form the quadratic and identify the noncentrality.

Answer

Under $P_{n}$ , $(Δ_{n}, Λ_{n}) d (Δ, Λ)$ with $Δ \sim N (0, I)$ and $Λ = ⟨ h, Δ ⟩ - \frac{1}{2} h^{T} I h$ , so $Cov (Δ, Λ) = Cov (Δ, ⟨ h, Δ ⟩) = I h$ . By the third lemma the alternative limit of $Δ_{n}$ is the null limit shifted in mean by $Cov (Δ, Λ) = I h$ : $Δ_{n} d N (I h, I)$ under $Q_{n}$ . Then $S_{n} = Δ_{n}^{T} I^{- 1} Δ_{n} d Z^{T} I^{- 1} Z$ with $Z \sim N (I h, I)$ . Writing $Y = I^{- 1/2} Z \sim N (I^{1/2} h, Id)$ , $S = Y^{T} Y = ∥ Y ∥^{2}$ is a sum of $k$ squared independent unit-variance normals with mean vector $I^{1/2} h$ , i.e. $χ_{k}^{2} (λ)$ with noncentrality $λ = ∥ I^{1/2} h ∥^{2} = h^{T} I h$ . Rubric: full credit for the shifted mean $I h$ , the whitening to $N (I^{1/2} h, Id)$ , and the noncentrality $h^{T} I h$ . This is the local power of the Wald/score/likelihood-ratio family of 45.04.09, obtained without ever computing the sampling distribution of $S_{n}$ under each alternative.

Advanced results Master

Contiguity is the change-of-measure backbone of the local theory: it certifies that limit statements proved under the null transport to drifting alternatives, and Le Cam's three lemmas are the precise transport rules. The first lemma reduces contiguity to mass conservation of the likelihood-ratio limit; the second supplies the asymptotically normal log-ratio that LAN produces; the third converts any joint null limit into an alternative limit by a Gaussian tilt. The downstream payoff is that local power — the limiting rejection probability of a test under a sequence of alternatives approaching the null at rate $n$ — is computed once from the null limit and a covariance, yielding the noncentral chi-squared limits of 45.04.09 uniformly across the Wald, score, and likelihood-ratio statistics.

Theorem 1 (Le Cam's first lemma; the contiguity characterisation). Under the hypotheses of the Key theorem, $Q_{n} ◃ P_{n}$ iff every subsequential limit $W$ of $d Q_{n} / d P_{n}$ under $P_{n}$ satisfies $E [W] = 1$ , iff every subsequential limit $V$ of $d P_{n} / d Q_{n}$ under $Q_{n}$ has $P (V = 0) = 0$ . Equivalently, in terms of the Hellinger affinity $ρ_{n} = E_{P_{n}} d Q_{n} / d P_{n}$ , no escaping mass is the condition that $ρ_{n}$ stays bounded away from $0$ along the relevant subsequence, the asymptotic refinement of the Kakutani affinity dichotomy 37.04.04 ^{[van der Vaart — Asymptotic Statistics]}.

Theorem 2 (Le Cam's second lemma; asymptotic normality of the LAN log-ratio). In a LAN sequence at $θ_{0}$ with central sequence $Δ_{n, θ_{0}} d N (0, I (θ_{0}))$ , the local log-likelihood ratio $Λ_{n} = lo g (d P_{θ_{0} + h / n}^{\otimes n} / d P_{θ_{0}}^{\otimes n})$ satisfies, under $P_{θ_{0}}^{\otimes n}$ ,

Λ_{n} d N (- \frac{1}{2} σ^{2}, σ^{2}), σ^{2} = h^{T} I (θ_{0}) h,

and jointly $(Δ_{n, θ_{0}}, Λ_{n}) d (Δ, ⟨ h, Δ ⟩ - \frac{1}{2} σ^{2})$ with $Δ \sim N (0, I (θ_{0}))$ . The mean-equals-minus-half-variance structure makes the alternatives contiguous to the null by Theorem 1, and the joint Gaussian limit is the input to the third lemma ^{[van der Vaart — Asymptotic Statistics]}.

Theorem 3 (Le Cam's third lemma; the shifted limit). Suppose under $P_{n}$ a sequence of statistics $T_{n}$ (valued in $R^{m}$ ) and the log-ratio $Λ_{n}$ satisfy

(T_{n}, Λ_{n}) d N ((μ - \frac{1}{2} σ^{2}), (Σ c^{T} c σ^{2})),

with $Q_{n} ◃ P_{n}$ . Then under $Q_{n}$ ,

T_{n} d N (μ + c, Σ) .

The covariance $Σ$ is preserved and the mean is shifted by the cross-covariance $c = Cov (T, Λ)$ of the statistic with the log-ratio in the joint null limit ^{[Le Cam, L. — Locally asymptotically normal families of distributions]}.

Theorem 4 (local power of asymptotically linear tests). Let a test reject when $T_{n} = a^{T} Δ_{n, θ_{0}} \to N (0, a^{T} I a)$ exceeds a null-calibrated cutoff. Under the contiguous alternative $θ_{0} + h / n$ , the third lemma gives $T_{n} d N (a^{T} I h, a^{T} I a)$ ; the limiting power is monotone in the standardised noncentrality $(a^{T} I h) / (a^{T} I a)^{1/2}$ , maximised over directions $a$ by $a \propto h$ (the efficient score direction), and the maximal local power is the Neyman-Pearson power for the limit Gaussian shift experiment ${N (h, I^{- 1})}$ of 45.04.06. The multidimensional quadratic statistic has the noncentral chi-squared limit $χ_{k}^{2} (h^{T} I h)$ of 45.04.09 ^{[Hájek, J. & Šidák, Z. — Theory of Rank Tests]}.

Theorem 5 (the general, possibly singular, first lemma). Drop $Q_{n} ≪ P_{n}$ . Write the Lebesgue decomposition $Q_{n} = Q_{n}^{a c} + Q_{n}^{⊥}$ with respect to $P_{n}$ and let $W_{n} = d Q_{n}^{a c} / d P_{n}$ . Then $Q_{n} ◃ P_{n}$ iff (i) every subsequential limit $W$ of $W_{n}$ under $P_{n}$ has $E [W] = 1$ and (ii) $Q_{n}^{⊥} (X_{n}) \to 0$ . Contiguity thus simultaneously controls the escaping mass of the absolutely continuous part and forces the singular part to vanish; the affinity form $ρ_{n} = E_{P_{n}} W_{n} \to 1$ (along the subsequence) encodes both ^{[Roussas, G. G. — Contiguity of Probability Measures: Some Applications in Statistics]}.

Synthesis. The foundational reason these lemmas cohere is that the likelihood ratio is a non-negative mean-one martingale-like object whose only pathologies are leakage to $0$ or $\infty$ , so contiguity, mass conservation of the limit, and uniform integrability of the ratios are one condition viewed through the negligible-set definition, the Fatou/mean-convergence inequality, and Prohorov tightness respectively. This is exactly the asymptotic incarnation of the Kakutani dichotomy 37.04.04: the Hellinger affinity that decided absolute continuity versus singularity there decides contiguity versus escape here, and the square root that linearised the product into a sum in Kakutani is the same root-likelihood that linearises the LAN log-ratio of 45.04.06. The third lemma is dual to the first: where the first rules out escaping mass so that the change-of-measure operator $f \mapsto E_{P_{n}} [f e^{Λ_{n}}]$ is asymptotically mass-preserving, the third computes its action on a jointly Gaussian statistic as a mean shift by the covariance $c$ , the Gaussian exponential tilt. Putting these together, the central insight is that the entire alternative-distribution theory of the chapter — local power, noncentral chi-squared limits, the asymptotic relative efficiency of competing tests — is the null-distribution theory transported by one covariance, so that no sampling distribution is ever recomputed under a moving alternative. The bridge is convergence of experiments: the local experiments converge to the Gaussian shift 45.04.06, contiguity is what makes that convergence a genuine transfer of risk and power, and the three lemmas are its operational form.

Full proof set Master

The first lemma and the Gaussian special case are proved in the Key theorem; the second and third lemmas and the local-power consequences are recorded here.

Proposition 1 (second lemma: LAN gives an asymptotically normal log-ratio). In a LAN sequence at $θ_{0}$ , $Λ_{n} d N (- \frac{1}{2} σ^{2}, σ^{2})$ under $P_{θ_{0}}^{\otimes n}$ , with $σ^{2} = h^{T} I (θ_{0}) h$ , and $(Δ_{n, θ_{0}}, Λ_{n})$ converges jointly to $(Δ, ⟨ h, Δ ⟩ - \frac{1}{2} σ^{2})$ , $Δ \sim N (0, I (θ_{0}))$ .

Proof. The LAN expansion 45.04.06 gives $Λ_{n} = ⟨ h, Δ_{n, θ_{0}} ⟩ - \frac{1}{2} h^{T} I h + o_{P_{n}} (1)$ . By the multivariate central limit theorem 37.03.02, $Δ_{n, θ_{0}} d Δ \sim N (0, I)$ , so the continuous-mapping theorem applied to the affine map $x \mapsto (x, ⟨ h, x ⟩ - \frac{1}{2} h^{T} I h)$ gives $(Δ_{n, θ_{0}}, ⟨ h, Δ_{n, θ_{0}} ⟩ - \frac{1}{2} h^{T} I h) d (Δ, ⟨ h, Δ ⟩ - \frac{1}{2} σ^{2})$ , and Slutsky absorbs the $o_{P_{n}} (1)$ . The second coordinate is normal with mean $- \frac{1}{2} σ^{2}$ and variance $Var (⟨ h, Δ ⟩) = h^{T} I h = σ^{2}$ . $□$

Proposition 2 (third lemma). If under $P_{n}$ , $(T_n, \Lambda_n) \xrightarrow{d} N\big((\mu, -\tfrac12\sigma^2), \begin{psmallmatrix}\Sigma & c\\ c^{\mathsf T} & \sigma^2\end{psmallmatrix}\big)$ and $Q_{n} ◃ P_{n}$ , then under $Q_{n}$ , $T_{n} d N (μ + c, Σ)$ .

Proof. By the first lemma the family ${e^{Λ_{n}}}$ is uniformly integrable under $P_{n}$ (mean-one convergence to $e^{Λ}$ with $E [e^{Λ}] = 1$ ). For bounded continuous $f$ on $R^{m}$ , the map $(t, λ) \mapsto f (t) e^{λ}$ is continuous and, by the UI of $e^{Λ_{n}}$ and the joint weak convergence, $E_{Q_{n}} f (T_{n}) = E_{P_{n}} [f (T_{n}) e^{Λ_{n}}] \to E [f (T) e^{Λ}]$ . It remains to identify the tilted law $ν (f) = E [f (T) e^{Λ}]$ . The joint density of $(T, Λ)$ is the multivariate normal with the stated mean and covariance; multiplying by $e^{λ}$ and integrating out $λ$ completes the square. Concretely the characteristic function is

\mathbb{E}[e^{\mathrm i u^{\mathsf T} T} e^{\Lambda}] = \exp\Big(\mathrm i u^{\mathsf T}\mu - \tfrac12\sigma^2 + \tfrac12\big(\mathrm i u, 1\big)\begin{psmallmatrix}\Sigma & c\\ c^{\mathsf T} & \sigma^2\end{psmallmatrix}\begin{psmallmatrix}\mathrm i u\\ 1\end{psmallmatrix}\Big),

and expanding the quadratic form gives exponent $i u^{T} μ - \frac{1}{2} σ^{2} - \frac{1}{2} u^{T} Σ u + i u^{T} c + \frac{1}{2} σ^{2} = i u^{T} (μ + c) - \frac{1}{2} u^{T} Σ u$ . This is the characteristic function of $N (μ + c, Σ)$ , and $ν$ is a probability law since $ν (1) = E [e^{Λ}] = 1$ . Hence under $Q_{n}$ , $T_{n} d N (μ + c, Σ)$ . $□$

Proposition 3 (local power of the score test). In a LAN family at $θ_{0}$ , the one-sided score test in direction $h_{0}$ based on $T_{n} = (h_{0}^{T} I h_{0})^{- 1/2} ⟨ h_{0}, Δ_{n, θ_{0}} ⟩$ has limiting power $1 - Φ (z_{1 - α} - δ)$ against $θ_{0} + h_{0} / n$ , with $δ = (h_{0}^{T} I h_{0})^{1/2}$ .

Proof. Set $σ^{2} = h_{0}^{T} I h_{0}$ . The pair $(T_{n}, Λ_{n})$ with $Λ_{n} = ⟨ h_{0}, Δ_{n} ⟩ - \frac{1}{2} σ^{2}$ has joint null limit $(T, Λ)$ Gaussian with $T \sim N (0, 1)$ , $Cov (T, Λ) = σ^{- 1} Cov (⟨ h_{0}, Δ ⟩, ⟨ h_{0}, Δ ⟩) = σ^{- 1} σ^{2} = σ = δ$ . By Proposition 2 the alternative limit of $T_{n}$ is $N (δ, 1)$ . Therefore $lim_{n} Q_{n} (T_{n} > z_{1 - α}) = P (N (δ, 1) > z_{1 - α}) = 1 - Φ (z_{1 - α} - δ)$ , increasing in $δ = (h_{0}^{T} I h_{0})^{1/2}$ . $□$

Proposition 4 (noncentral chi-squared local power). For the quadratic score statistic $S_{n} = Δ_{n, θ_{0}}^{T} I (θ_{0})^{- 1} Δ_{n, θ_{0}}$ with null limit $χ_{k}^{2}$ , the contiguous-alternative limit is $χ_{k}^{2} (λ)$ with $λ = h^{T} I (θ_{0}) h$ .

Proof. The joint null limit of $(Δ_{n}, Λ_{n})$ is Gaussian with $Cov (Δ, Λ) = Cov (Δ, ⟨ h, Δ ⟩) = I h$ . By Proposition 2 applied to $T_{n} = Δ_{n}$ , under $Q_{n}$ , $Δ_{n} d N (I h, I)$ . Setting $Y = I^{- 1/2} Δ_{n}$ 's limit $\sim N (I^{1/2} h, Id)$ , $S_{n} d ∥ Y ∥^{2} \sim χ_{k}^{2} (λ)$ with $λ = ∥ I^{1/2} h ∥^{2} = h^{T} I h$ , since a sum of $k$ squared independent unit-variance normals with mean vector $m$ is noncentral chi-squared with noncentrality $∥ m ∥^{2}$ . $□$

Proposition 5 (contiguity from a bounded limiting Hellinger affinity). For sequences $P_{n}, Q_{n}$ with $ρ_{n} = E_{P_{n}} d Q_{n} / d P_{n}$ , if $lim inf_{n} ρ_{n} > 0$ then $Q_{n} ◃ P_{n}$ .

Proof. Let $W_{n} = d Q_{n} / d P_{n}$ and suppose along a subsequence $W_{n} d W$ under $P_{n}$ (tightness from $E_{P_{n}} W_{n} = 1$ ). The map $w \mapsto w$ is continuous and bounded by $1 + w$ , and ${W_{n}}$ has bounded means, so $W_{n}$ is uniformly integrable; hence $E [W] = lim_{n} E_{P_{n}} W_{n} = lim_{n} ρ_{n} \geq δ > 0$ . If mass had escaped, $E [W] < 1$ would force, by the Cauchy-Schwarz relation $E [W] \leq E [W] P (W > 0)$ together with the escaping-mass accounting, a strictly smaller affinity; the quantitative bound $1 - ρ_{n} \leq h^{2} (P_{n}, Q_{n})$ and $∣ Q_{n} (A_{n}) - P_{n} (A_{n}) ∣ \leq 2 h (P_{n}, Q_{n}) \leq 2 (1 - δ + o (1))$ together with the first lemma give $E [W] = 1$ , hence contiguity. $□$

Connections Master

The local asymptotic normality of 45.04.06 is the source of every concrete contiguity statement in this chapter: the LAN log-likelihood expansion is exactly the hypothesis under which the second lemma produces the $N (- \frac{1}{2} σ^{2}, σ^{2})$ log-ratio, the first lemma then certifies mutual contiguity of the local alternatives, and the central sequence $Δ_{n, θ_{0}}$ is the statistic whose joint limit with the log-ratio the third lemma tilts; contiguity is what upgrades the convergence of local experiments there into a genuine transfer of risk and power.

Kakutani's theorem on product martingales and absolute continuity 37.04.04 is the finite- $n$ ancestor of the first lemma: the Hellinger affinity $\prod_{k} a_{k}$ that decides absolute continuity versus singularity of infinite product measures is the same affinity $ρ_{n} = E_{P_{n}} d Q_{n} / d P_{n}$ whose boundedness away from zero decides contiguity, and the square-root linearisation that turns the product martingale into an $L^{2}$ -bounded object is the root-likelihood that linearises the LAN log-ratio; contiguity is the moving-target, drifting-alternative refinement of the static Kakutani dichotomy.

The chi-squared asymptotics of the Wald, score, and likelihood-ratio tests 45.04.09 are the principal payoff of the third lemma: the noncentral chi-squared limit $χ_{k}^{2} (h^{T} I h)$ under contiguous alternatives, and hence the local power and asymptotic relative efficiency of these tests, are computed by the single covariance shift of Proposition 4 rather than by re-deriving each statistic's distribution under each moving alternative, so the entire local-power table of that unit is a corollary of contiguity.

The consistency and modes-of-convergence machinery of 45.04.01 supplies the $o_{P}$ calculus, the continuous-mapping and Slutsky lemmas, and the Prohorov tightness that the first lemma's uniform-integrability argument rests on; contiguity is the change-of-measure layer built directly on top of those convergence modes.

The delta method of 45.04.02 composes with the third lemma: a smooth transformation of a statistic whose null limit is known has its alternative limit obtained by first tilting (third lemma) and then linearising (delta method), so the local distribution of a plug-in functional under a contiguous alternative is again a translated Gaussian, with the translation propagated through the derivative of the functional.

Historical & philosophical context Master

The notion of contiguity is due to Lucien Le Cam, introduced in his 1960 monograph Locally asymptotically normal families of distributions ^{[Le Cam, L. — Locally asymptotically normal families of distributions]}, where the first and third lemmas appear as tools for transferring asymptotic statements between a null hypothesis and a sequence of local alternatives. Le Cam isolated the property that makes such transfer possible — that the alternatives never place asymptotically non-negligible mass where the null places vanishing mass — and recognised it as the sequence-level analogue of absolute continuity, decided by the limiting behaviour of the likelihood ratio. The connection to the Hellinger affinity, and through it to Kakutani's 1948 dichotomy for product measures ^{[van der Vaart — Asymptotic Statistics]}, situates contiguity within the older measure-theoretic theory of equivalence and singularity of infinite products.

The three lemmas were systematised and put to work by Jaroslav Hájek and Zbyněk Šidák in their 1967 Theory of Rank Tests ^{[Hájek, J. & Šidák, Z. — Theory of Rank Tests]}, where contiguity is the engine that yields the asymptotic distributions and local power of rank statistics under contiguous location-shift alternatives — distributions that are otherwise intractable. George Roussas's 1972 monograph extended contiguity to dependent and Markov data, giving the general possibly-singular first lemma and its applications ^{[Roussas, G. G. — Contiguity of Probability Measures: Some Applications in Statistics]}. The modern textbook synthesis, in which the first lemma is read off the LAN log-ratio and the third lemma is a one-line Gaussian tilt, follows van der Vaart's Asymptotic Statistics, where contiguity is presented as the bridge between the null-distribution theory and all alternative-distribution computations of the local theory.

Bibliography Master

@book{vanderVaart1998,
  author    = {van der Vaart, Aad W.},
  title     = {Asymptotic Statistics},
  publisher = {Cambridge University Press},
  series    = {Cambridge Series in Statistical and Probabilistic Mathematics},
  year      = {1998}
}

@book{LeCamYang2000,
  author    = {Le Cam, Lucien and Yang, Grace Lo},
  title     = {Asymptotics in Statistics: Some Basic Concepts},
  edition   = {2},
  series    = {Springer Series in Statistics},
  publisher = {Springer},
  year      = {2000}
}

@article{LeCam1960,
  author  = {Le Cam, Lucien},
  title   = {Locally asymptotically normal families of distributions},
  journal = {University of California Publications in Statistics},
  volume  = {3},
  year    = {1960},
  pages   = {37--98}
}

@book{HajekSidak1967,
  author    = {H\'ajek, Jaroslav and \v{S}id\'ak, Zbyn\v{e}k},
  title     = {Theory of Rank Tests},
  publisher = {Academic Press},
  year      = {1967}
}

@book{HajekSidakSen1999,
  author    = {H\'ajek, Jaroslav and \v{S}id\'ak, Zbyn\v{e}k and Sen, Pranab K.},
  title     = {Theory of Rank Tests},
  edition   = {2},
  publisher = {Academic Press},
  year      = {1999}
}

@book{Roussas1972,
  author    = {Roussas, George G.},
  title     = {Contiguity of Probability Measures: Some Applications in Statistics},
  publisher = {Cambridge University Press},
  year      = {1972}
}

@article{Kakutani1948,
  author  = {Kakutani, Shizuo},
  title   = {On equivalence of infinite product measures},
  journal = {Annals of Mathematics},
  volume  = {49},
  number  = {1},
  pages   = {214--224},
  year    = {1948}
}

Prerequisites

45.04.06

Tier anchors

beginner: van der Vaart 1998 Asymptotic Statistics (Cambridge) Ch. 6 (contiguity as the asymptotic version of absolute continuity, the opening informal discussion of why one wants to transfer limit statements from a null to a nearby alternative without redoing the calculation); Le Cam & Yang 2000 Asymptotics in Statistics 2e (Springer) Ch. 3 (contiguity, expository opening on the change-of-measure idea)
intermediate: van der Vaart 1998 Asymptotic Statistics (Cambridge) §6.1-6.2 (definition of contiguity, Le Cam's first lemma characterising contiguity through the limit law of the likelihood ratio, and Le Cam's third lemma for the shifted limit of a statistic under a contiguous alternative); Hájek & Šidák 1967 Theory of Rank Tests (Academic Press) Ch. VI (contiguity and the LeCam lemmas as the engine for the asymptotic distribution theory of rank statistics under alternatives)
master: van der Vaart 1998 Asymptotic Statistics (Cambridge) Ch. 6 (contiguity §6.1, the first, second, and third lemmas §6.2-6.4, applications to local power §6.5) and Ch. 7 (the LAN log-likelihood expansion that supplies the asymptotically normal log-ratio); Le Cam & Yang 2000 Asymptotics in Statistics 2e (Springer) Ch. 3 (contiguity, the relation to Hellinger affinities and the Kakutani dichotomy) and Ch. 6 (the LAN consequences); Le Cam 1960 Locally asymptotically normal families (Univ. Calif. Publ. Statist. 3) §§3-5 (the original first and third lemmas); Hájek, Šidák & Sen 1999 Theory of Rank Tests 2e (Academic Press) Ch. 7 (contiguity machinery for local power of rank tests)

References

van der Vaart — Asymptotic Statistics · Cambridge University Press, Cambridge Series in Statistical and Probabilistic Mathematics, 1998; Ch. 6 (contiguity §6.1; Le Cam's first lemma characterising contiguity via the limit law of the likelihood ratio §6.2; the second lemma / LAN log-likelihood expansion as the source of an asymptotically normal log-ratio §6.3; Le Cam's third lemma, the shifted limit of a statistic under a contiguous alternative §6.4; local power of tests §6.5) and Ch. 7 (the LAN expansion §7.2-7.4)
Le Cam, L. & Yang, G. L. — Asymptotics in Statistics: Some Basic Concepts (2nd ed.) · Springer Series in Statistics, 2000; Ch. 3 (contiguity, mutual contiguity, the relation to Hellinger affinities and the Kakutani absolute-continuity-versus-singularity dichotomy, Le Cam's first and third lemmas) and Ch. 6 (the local-asymptotic-normality consequences and local power)
Le Cam, L. — Locally asymptotically normal families of distributions · University of California Publications in Statistics 3 (1960), 37-98 (the original definition of contiguity, the first lemma characterising it through the likelihood-ratio limit, and the third lemma computing the limit law of a statistic under a contiguous sequence of alternatives)
Hájek, J. & Šidák, Z. — Theory of Rank Tests · Academic Press, 1967; Ch. VI (LeCam's contiguity lemmas systematised and applied to derive the asymptotic distribution and local power of rank statistics under contiguous location-shift alternatives)
Roussas, G. G. — Contiguity of Probability Measures: Some Applications in Statistics · Cambridge University Press, 1972; Ch. 1-2 (a book-length treatment of contiguity, the first and third lemmas, and their use in asymptotic inference for dependent and Markov data)

Estimated time

beginner: 20m
intermediate: 55m
master: 95m