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In general, no. For example, suppose $A_k$ are iid $N(0,1)$ and $B_k$ are also iid $N(0,1)$ and $X_{2k}=A_k+B_k$, $X_{2k+1}= A_k-B_k$. Then the $B_k$ all cancel and $$\bar Y_{2k}= \frac{2}{\sqrt{2k}}\sum_k A_k\sim N(0, 2)$$ Suppose $W_k$ is binary with values 0 and 2. Instead of the $B_k$ all cancelling, about half of them will be present once, and the ...


2

The claimed equation is not true. Or, $\sqrt{n}(\widehat{\beta}-\beta_0)$ and $\sqrt{n}(\widetilde{\beta}-\beta_0)$ is not asymptotically equivalent. To see it, note that $\frac{1}{n}\sum_{i=1}^{n}\widehat{T}_{i}\widehat{T}_{i}^{\top}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}(X_i=1)\begin{bmatrix}1&\widehat{E}(D|X_i=1)\\ \widehat{E}(D|X_i=1)&(\widehat{E}(D|...


1

Posting questions on CV has this magic effect many times: a little after you post them, you find the answer. ANSWER: both expressions are correct, and the expression in Theorem B holds also for the diagonal variance terms. The catch? ...In the variance-covariance matrix expression, we realize that the numbering of the variables starts from $2$, while the $i,...


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"Then the sequence of partial sums $\{\sum_{r=1}^{a_n-1} c(kr)\}_{n\in\mathbb{N}}$ is a subsequence of $\{\sum_{r=1}^{n-1} c(r)\}_{n\in\mathbb{N}}$"---this statement is not correct. Your final result is correct. A stationary AR(1) process has autocovariance function $\gamma(r) = \rho^r$ (using more standard notation $\gamma$ instead of $c$) When you $k$-...


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