Independent-subject terminal influence limits

Given: Assume AC. The subjects $O_i$ are IID with law $P$; the measurable influence function $\varphi$ is centered and square-integrable; $\mathrm{\Sigma }=E_P[\varphi {\varphi }^T]$; and the estimator has the displayed expansion with $\sqrt{n}{r}_n{\to }_{P}0$. Prove the displayed distributional limit.

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

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

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

Asymptotically linear estimators and influence functions: For an IID sample, an asymptotically linear estimator has expansion ${\widehat{\psi }}_n-\psi (P)=(P_n-P)\varphi +o_P(n^{-1\mathrm{/}2})$, with a mean-zero square-integrable influence function; this definition alone does not assert asymptotic normality.

Covariance matrix of a finite-dimensional random vector: For a finite-dimensional square-integrable vector $X$ with mean $\mu$, $\mathrm{Cov}(X)=E[(X-\mu )(X-\mu {)}^T]$.

Multivariate iid central limit theorem: Under AC, IID centered ${\mathbb{R}}^d$ vectors with finite second moment and covariance $\mathrm{\Sigma }$ have normalized sums converging to $N_d(0,\mathrm{\Sigma })$; $\mathrm{\Sigma }$ may be singular.

Stochastic order and convergence notation for finite-dimensional random vectors: For vectors, $R_n=o_P(1)$ means $P(\mathrm{\parallel }{R}_n{\mathrm{\parallel }}_2>\epsilon )\to 0$ for every $\epsilon >0$.

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 $(X,c)$.

Given: Assume AC, fixed finite $G\ge 2$ and $d\ge 1$, deterministic sample sizes with $n={\sum }_g{n}_g$ and $n_g\mathrm{/}n\to {\rho }_g\in (0,1)$, independent subject families, centered square-integrable measurable ${\varphi }_g$ with covariance ${\mathrm{\Sigma }}_g$, and the displayed estimator expansion with $R_n=o_P(1)$. Prove the displayed limit.

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

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

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.

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

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

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 $(X,c)$.

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

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

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

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

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

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.

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 $u^{T}Vu$.

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.

Stochastic order and convergence notation for finite-dimensional random vectors: For vectors, $R_n=o_P(1)$ means $P(\mathrm{\parallel }{R}_n{\mathrm{\parallel }}_2>\epsilon )\to 0$ for every $\epsilon >0$.