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Independent-group terminal influence limit

Statement

Assume the Axiom of Choice. Fix a finite $G \geq 2$ and $d \geq 1$ . For each total sample size $n$ , let $n_{1}, \dots, n_{G}$ be positive deterministic integers with $\sum_{g = 1}^{G} n_{g} = n$ , and suppose $n_{g} / n \to ρ_{g} \in (0, 1)$ with $\sum_{g = 1}^{G} ρ_{g} = 1$ . For group $g$ , let $O_{g 1}, O_{g 2}, \dots$ be IID with law $P_{g}$ on a measurable space; assume the subject families in distinct groups are independent. Let $ϕ_{g}$ be measurable with $E_{P_{g}} ϕ_{g} = 0$ , $E_{P_{g}} ∥ ϕ_{g} ∥_{2}^{2} < \infty$ , and $Σ_{g} = E_{P_{g}} [ϕ_{g} ϕ_{g}^{T}]$ . If a measurable estimator of fixed $θ \in R^{d}$ has the separately established expansion, with remainder on the same sample probability space as its group sums,

$\sqrt{n} ({\hat{θ}}_{n} - θ) = \sum_{g = 1}^{G} \sqrt{\frac{n}{n_{g}}} \frac{1}{\sqrt{n_{g}}} \sum_{i = 1}^{n_{g}} ϕ_{g} (O_{g i}) + R_{n}, R_{n} = o_{P} (1),$

then

$\sqrt{n} ({\hat{θ}}_{n} - θ) \Rightarrow N_{d} ⁣ (0, \sum_{g = 1}^{G} ρ_{g}^{- 1} Σ_{g}) .$

The group count $G$ is fixed; the laws $P_{g}$ may differ. The covariance may be singular. This is a terminal finite-dimensional result: it neither proves the estimator expansion nor supplies a continuous-time or functional martingale CLT.

Facts & Assumptions

Given: Assume AC, fixed finite $G \geq 2$ and $d \geq 1$ , deterministic sample sizes with $n = \sum_{g} n_{g}$ and $n_{g} / n \to ρ_{g} \in (0, 1)$ , independent subject families, centered square-integrable measurable $ϕ_{g}$ with covariance $Σ_{g}$ , and the displayed estimator expansion with $R_{n} = o_{P} (1)$ . Prove the displayed limit.

[A1]

The Axiom of Choice: Assume the Axiom of Choice (AC).

[F1]

Identical distribution and IID families: Within each group, IID means a common law and mutual independence of the whole family.

[F2]

Independent families of event classes: For every finite selection of distinct independent indices and one event from each selected class, the probability of their intersection is the product of their probabilities.

[F3]

Covariance matrix of a finite-dimensional random vector: For a finite-dimensional square-integrable vector $X$ with mean $μ$ , $Cov (X) = E [(X - μ) (X - μ)^{T}]$ and is positive semidefinite.

[F4]

Multivariate iid central limit theorem: Under AC, IID centered $R^{d}$ vectors with finite second moment and covariance $Σ$ have normalized sums converging to $N_{d} (0, Σ)$ , including singular $Σ$ .

[F5]

Vector Slutsky theorem: If $X_{n} \Rightarrow X$ and $Y_{n} \to_{P} c$ , then every continuous function of $(X_{n}, Y_{n})$ converges in distribution to the same function of $(X, c)$ .

[F6]

Continuous mapping theorem: A continuous map sends a weakly convergent random vector to the corresponding weak limit.

[F7]

Characteristic function of a real random variable: The characteristic function is $t \mapsto E [e^{i t X}]$ .

[F8]

Levy continuity theorem forward direction: Weak convergence of real laws implies pointwise convergence of their characteristic functions.

[F9]

Characteristic functions under affine maps and independent sums: For real $a, b, t$ , $φ_{a X + b} (t) = e^{i t b} φ_{X} (a t)$ ; for a finite mutually independent family of real random variables, the characteristic function of its sum is the product of the individual characteristic functions.

[F10]

Characteristic function of a normal law: A centered real normal law of variance $v \geq 0$ has characteristic function $t \mapsto e^{- v t^{2} / 2}$ , including $v = 0$ .

[F11]

Characteristic function criterion for weak convergence: Under AC, pointwise convergence of characteristic functions to that of a specified real probability law implies weak convergence to that law.

[F12]

Multivariate normal law, including singular covariance: Under AC, $N_{d} (0, V)$ exists for every positive semidefinite $V$ and each projection $u^{T} X$ is normal with variance $u^{T} V u$ .

[F13]

Cramer wold device: Under AC, convergence of every scalar projection to the projections of a specified finite-dimensional law implies vector weak convergence to that law.

[F14]

Stochastic order and convergence notation for finite-dimensional random vectors: For vectors, $R_{n} = o_{P} (1)$ means $P (∥ R_{n} ∥_{2} > ε) \to 0$ for every $ε > 0$ .

Proof

1.1A1F1F2F3F4given

For each group, measurable preimages show that $ϕ_{g} (O_{g i})$ are IID; [F2] applied to preimages of finite event selections also shows independence across groups. Their means and covariances are $0$ and $Σ_{g}$ by assumption and [F3]. Since $ρ_{g} > 0$ and $n \to \infty$ , $n_{g} \to \infty$ . Applying [F4] to each of the finitely many groups gives $U_{g, n} : = n_{g}^{- 1 / 2} \sum_{i = 1}^{n_{g}} ϕ_{g} (O_{g i}) \Rightarrow Z_{g} \sim N_{d} (0, Σ_{g})$ .

2.1A1F5F6F9F10F12givenstep 1.1

Put $a_{g, n} = \sqrt{n / n_{g}}$ , so $a_{g, n} \to ρ_{g}^{- 1 / 2}$ . Applying [F5] to the continuous map $(x, a) \mapsto a x$ gives $a_{g, n} U_{g, n} \Rightarrow ρ_{g}^{- 1 / 2} Z_{g}$ . For any fixed $u \in R^{d}$ , [F6] gives convergence of the projections; by [F12], $u^{T} Z_{g}$ is normal, and the affine characteristic-function identity [F9] with [F10] identifies its scaled law as $N (0, ρ_{g}^{- 1} u^{T} Σ_{g} u)$ , including zero projected variance.

3.1A1F2F3F7F8F9F10F11F12step 2.1

Let $V = \sum_{g} ρ_{g}^{- 1} Σ_{g}$ . By [F3], $u^{T} V u = \sum_{g} ρ_{g}^{- 1} u^{T} Σ_{g} u \geq 0$ for every $u$ , so $V$ is positive semidefinite; under [A1], [F12] supplies $Z \sim N_{d} (0, V)$ . Set $W_{n} = \sum_{g} a_{g, n} U_{g, n}$ . Fix $u \in R^{d}$ and any frequency $t \in R$ . The scalar terms $u^{T} a_{g, n} U_{g, n}$ are independent across $g$ by [F2]. By the characteristic-function definition [F7], weak-convergence continuity [F8], independent-sum product identity [F9], and normal formula [F10], step 2.1 gives pointwise convergence of the characteristic function of $u^{T} W_{n}$ to $\prod_{g = 1}^{G} \exp (- t^{2} u^{T} Σ_{g} u / (2 ρ_{g})) = \exp (- t^{2} u^{T} V u / 2)$ . This is the characteristic function of $u^{T} Z$ by [F12] and [F10]. The criterion [F11] gives $u^{T} W_{n} \Rightarrow u^{T} Z$ .

4.1A1F13step 3.1

Since step 3.1 holds for every $u \in R^{d}$ and the specified target is $Z$ , [F13] gives $W_{n} \Rightarrow Z \sim N_{d} (0, V)$ .

5.1F5F14step 4.1∎

The expansion in the statement is $W_{n} + R_{n}$ . By [F14], $R_{n} = o_{P} (1)$ is convergence in probability to the zero vector. Apply [F5] with the continuous addition map to step 4.1 to obtain the claimed estimator limit. This proof derives the group-sum limit but does not establish any consumer's expansion.

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Sources

Sourav Chatterjee, Stats 310A (Math 230A) Lecture Notes, §§8.5.3, 8.8–8.9, PDF pp. 104–107 / printed pp. 100–103 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., §3.10, Theorems 3.10.5–3.10.7, printed pp. 200–201 (standard reference, not scraped)
Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 16, PDF pp. 9–12 (standard reference, not scraped)