Drawing Samples from a Multivariate Normal distribution

Question

I have a quick question regarding generating samples from a multivariate normal distribution. When drawing samples from a standard normal distribution we can assume that the samples are i.i.d. However when drawing samples from a MVN with a co variance matrix which is not the identity matrix may we assume that they are i.i.d?

I am aware that the co variance matrix may highlight correlations between samples so I'm unsure if this nullifies the i.i.d property.Any input/explanation would be greatly appreciated.

Thanks in advance

Answer 1

Let us say that $(X_1, X_2) \sim BVN(\mu, \Sigma)$, where $$ \Sigma= \begin{pmatrix} \sigma^2_X & \rho\sigma_X\sigma_Y\\ \rho\sigma_Y\sigma_X & \sigma_Y^2 \end{pmatrix}, $$ for $\rho \neq 0$ clearly $X_1$ and $X_2$ are dependent variables with a known dependence structure, however the series $\{(x_{1i}, x_{2i})\}_{i=1}^n$ of random draws from it can be viewed as i.i.d. In other words, the i.i.d property holds for the random vectors that you draw and not for elements of the same vector.