I was looking at the Gauss-Markov theorem and its respective conditions but was confused by the notation.
From my understanding, the Gauss-Markov theorem deals with the error term denoted as $\epsilon$ and a few of the key portions of the theorem are as follows:
They have an arithmetic mean of zero: $E[\epsilon_i] = 0$
They have the same finite variance: Var$(\epsilon_i) = \sigma^2 < \infty$
And finally, they are not correlated: Cov$(\epsilon_i, \epsilon_j) = 0, \forall i\neq j$
My question is, if it is the case that the error term, $\epsilon$, is an $N$ dimensional vector, how could it be possible to take the arithmetic mean, variance, and covariance of a singular index of $\epsilon$ if it would be a scalar?
I am maybe misreading the notation but was unable to find any resources regarding the computation and would appreciate any help.
Thanks!