# Questions tagged [asymptotic-covariance]

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### Convergence in probability with double limits

Suppose you have a sequence of random variables $\left\lbrace X_{i}\right\rbrace_{i=1,...,n}$ which converges in probability to a random variable $X$, shown by $X_n \ \xrightarrow{p}\ X$ as n goes ...
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### Linear regression with one generated regressor

Suppose I have the regression model: $Y_i=T^{\top}_{i}\beta_0+e_{i}$ with $E(e_i|X_i)=0$, where we have two regressors $X_i,\ E(D|X_{i})$ so that $T^{\top}_{i}=[X_i,\ E(D|X_{i})]$. $X_{i}$ is a ...
46 views

### Variance of sample moments - clarification on Serfling (1980)

In "Serfling, R. J. (1980). Approximation theorems of mathematical statistics", we read In Theorem A, as one suspects, $k=1,2,...$, indicating the integer-moments, while $n$ is the sample ...
143 views

### variance of an autoregressive process

Let $\{x_t\}_{t\in\mathbb{N}}$ be a zero mean strictly stationary sequence of random variables and $c:\mathbb{N}\to\mathbb{R}$ the (auto)covariance function. If the process follows the AR(1) model ...
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### Do these two random variables have the same asymptotic distribution?

Let $\{X_k\}$ be a sequence of dependent random variables with mean 0. Define $\bar{Y}_k = \frac{1}{\sqrt k}\sum_{i=1}^k X_i$. Let $\{W_k\}$ be a sequence of i.i.d. random variables with mean 1 and ...
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### How to proof the asymptotic properties of the penalized spline estimator using asymptotic notations?

Please could someone proof how the Average Mean Squared Error of penalized spline estimator is given as \begin{eqnarray} AMSE(\hat{l} )=\...
50 views

### What is the sample form of the following variance

If $x_1, \ldots, x_n \in \mathbb{R}^p$ i.i.d. a specific distribution. denote: \begin{equation*} \mathbf{X} = \left( \begin{array}{c} x_1^{\top} \\ \cdots \\ x_n^{\top} ...