In the Wikipedia entry on the multivariate normal distribution, it says that one definition
is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.
However, since it's also true that any linear combination of normally distributed variables is itself normal, does this mean that any vector of univariate random normals is itself multivariate normal? Is there ever any situation where a vector of random normals is not multivariate normal?
What I should have said was "any linear combination of independent normally distributed variables is itself normal." All the answers below are good examples of variables that are not independent and thus not multivariate normal. So I should rephrase my question to be: is there ever any situation where a vector of independent random normals is not multivariate normal? I'm leaving the headline as is, to reflect history and the nature of the answers to this question, but I will alter it if you think I should. Sorry for any confusion I may have caused.