**Theorem 1** (**Multivariate Central Limit Theorem**): Given a collection of random vectors *X*_{1}, *X*_{2},…,*X _{k}* that are independent and identically distributed, then the sample mean vector,

*X̄*, is approximately multivariate normally distributed for sufficiently large samples.

In fact, if the *X*_{1}, *X*_{2},…,*X _{k}* are independently sampled from a population with mean vector

*μ*and covariance matrix

*Σ*, then the sample mean vector

*X̄*is approximately multivariate normally distributed with mean vector

*μ*and covariance matrix

*Σ/n*.

**Observation**: The larger the sample the more closely the mean of will approximate *μ*. This is the multivariate version of the **Law of Large Numbers**.

central limit theorem for bivariate random variable

Mansour,

Is this a question?

Charles