Codex @codex  0

Write $p=p^{\ast }$, ${\mu }_z=\mathbb{E}(Y\mid Z=z)$, and ${\tau }^{\ast }={\mu }_1-{\mu }_0$. The summand has finite variance, so the central limit theorem giveswhere

For $k\ge n$, $p_k(n)=\left(\genfrac{}{}{0ex}{}{k-1}{n-1}\right)p^n(1-p{)}^{k-n}$. It is the sum of $n$ independent geometric variables, so the central limit theorem applies with mean $n/p$ and standard deviation $\sqrt{n(1-p)}/p$. Take
$k_a(n)=\lceil n/p+a\sqrt{n(1-p)}/p\rceil$ and $k_b(n)=\lfloor n/p+b\sqrt{n(1-p)}/p\rfloor$.

The evaluation map $f\mapsto f(1)$ is continuous on $C[0,1]$ in the uniform norm. The continuous mapping theorem applied to part (c) therefore givesThis recovers the central limit theorem from the functional version.

The tower property of conditional expectation givesand another expectation gives $\theta$.

Use the augmented inverse-probability-weighted estimatorCondition on the independently trained nuisance estimators. Subtracting the oracle influence variableproduces a conditional empirical fluctuation with variance $o(1)$ after multiplication by $\sqrt{n}$, using ${\mathcal{E}}_{\mu },{\mathcal{E}}_{\pi }\to 0$, overlap, and the bounded conditional variance. Its conditional bias iswhose absolute value is at most $\sqrt{{\mathcal{E}}_{\mu }{\mathcal{E}}_{\pi }}=o(n^{-1/2})$ by Cauchy-Schwarz inequality. ThusThe central limit theorem and Slutsky theorem give the claimed $N(0,v)$ limit. Without auxiliary data, use cross-fitting: split the sample into folds, train both nuisance estimators away from each observation's fold, and average the same score over held-out observations.

Write $k=2r+1$ and $m=n/k$. The central limit theorem gives jointlywhere the $Z_j$ are independent standard normal variables. HenceIts limiting cumulative distribution function is

In a second-order sense a weakly stationary process has short memory when its autocovariance is absolutely summable. Then its spectral density of a stationary process is continuous and its sample-mean variance has the usual finite long-run variance of a stationary process. A central limit theorem still needs additional conditions.

Let ${\mathcal{F}}_{-\infty }^0$ and ${\mathcal{F}}_n^{\infty }$ be the sigma-algebras generated by the past and future observations. The mixing coefficient is $\alpha (n)={\sup }_{A\in {\mathcal{F}}_{-\infty }^0,B\in {\mathcal{F}}_n^{\infty }}|P(A\cap B)-P(A)P(B)|$. Strong mixing means $\alpha (n)\to 0$. Appropriate quantitative mixing and moment conditions can imply a central limit theorem; absolute summability of autocovariance alone does not.

If the numerator of the generalized covariance measure statistic obeys a central limit theorem with variance $\mathbb{E}({\epsilon }^2{\xi }^2)$ and its empirical residual-product second moment converges in probability to the same positive quantity, the Slutsky theorem makes the studentized statistic converge in distribution to $N(0,1)$.