When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero.
I am interested in a generalisation of this test to the case where instead a sample of numbers we have a sample of vectors. In other words, we sample from a multivariate distribution. For example, we have 1000 three dimensional vectors (which can be represented as a 1000 by 3 matrix).
My null hypothesis is that mean of this distribution is zero. The test have to take into account that the components of the vectors are not independent (correlated). Do such a test exist?