Multivariate Central Limit Theorem

Theorem 1 (Multivariate Central Limit Theorem): Given a collection of random vectors X1, X2,…,Xk that are independent and identically distributed, then the sample mean vector, , is approximately multivariate normally distributed for sufficiently large samples.

In fact, if the X1, X2,…,Xk are independently sampled from a population with mean vector μ and covariance matrix Σ, then the sample mean vector 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.

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