Central Limit Theorem
As observed in Other Multivariate Normal Distribution Properties
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, X̄, 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 X̄ is approximately multivariate normally distributed with mean vector μ and covariance matrix Σ/n.
Law of Large Numbers
The larger the sample the more closely the sample mean vector X̄ will approximate μ. This is the multivariate version of the Law of Large Numbers.
Reference
Rencher, A.C. (2002) Methods of multivariate analysis (2nd Ed). Wiley-Interscience, New York.
https://www.ipen.br/biblioteca/slr/cel/0241
what is considered a sufficiently large sample?
Hello Masoud,
For the univariate CLT, a sample of 30 is usually stated, although other sizes are also claimed. I don’t know what the estimate is for the multivariate CLT.
Charles
What do you mean by approximately multivariate normally distributed?
central limit theorem for bivariate random variable
Mansour,
Is this a question?
Charles