# Asymptotic covariance matrix of $\bar{\pmb x}$

In a text I'm reading it says that we define

\begin{align} \bar{\pmb x}= \begin{bmatrix}\bar x_1 \\ \bar x_2 \end{bmatrix} \end{align} And then immediately says the asymptotic covariance matrix for $$\bar{\pmb x}$$ is \begin{align} \pmb V = N E\left[(\bar{\pmb x} - \pmb \mu)(\bar{\pmb x} - \pmb \mu)' \right] \end{align}

But I'm having a hard time seeing that. I'm comfortable with the covariance matrix for $$\pmb x$$ being $$E[(\pmb x - \pmb \mu)(\pmb x - \pmb \mu)']$$ but for some reason I'm having a hard time seeing how $$N$$ comes into the picture even though I know it gets there via the CLT.

I know that the CLT says $$\bar x \sim N(\mu, \frac{\sigma^2}{N})$$ but still having a hard time getting from there to the definition of the asymptotic covariance matrix.

## Distributional Assumptions

$$x_i$$ is distributed multi-variate normal, independently, and i.i.d.

## Solution

It makes sense now, the variance isn't for $$\pmb x$$ but for $$\pmb{\bar x}$$. Therefore we have:

$$\bar x \sim N(\mu, \sigma^2_{\bar x})$$ which when we demean and multiply by $$\sqrt{N}$$ we have $$\sqrt{N}(\bar x-\mu) \sim N(0, N\sigma^2_{\bar x})$$.

Asymptotically, $$\bar{x} - \mu$$ goes to zero. So the asymptotical covariance of $$\bar{x}$$ refers to that of $$\sqrt{n}(\bar{x} - \mu)$$, which is $$V$$, when we assume all the regularity conditions of CLT are met.
By the way, we do not really need the distributional assumption on $$x$$.
$$V = Var(\sqrt{n}(\bar{x} - \mu)) = n Var(\bar{x}) = n E (\bar{x}-\mu)^2$$ which is what you have in your textbook.
In the event that $$x_t$$ are i.i.d, then $$n Var(\bar{x}) = Var(x)$$. Maybe this is what confused you? The textbook is written down without assuming i.i.d.
• Okay, do you think you could illustrate mathematically why we have $N$ out front for the the variance? $\bar x \sim N(\mu, \sigma^2/N$ and running the transformation described gives us $\sqrt{N}(\bar x - \mu) \sim N(0, \sigma^2)$ which still isn't $N\sigma^2$ Commented Aug 12, 2021 at 16:27