I don't know the context, what you mean with "a correlated complex [...] Gaussian distribution" and there is a missing assumption. You say that the epsilon are normally distributed. Suppose that they are also independently (this is the missing assumption) distributed. Then any linear combination of independent normal random variables (rv) is normal with a variance equal to a linear combination of the variances, but with weights equal to the squares of the initial weights. This is also true, but with different weights, if your rv have collectively a multivariate normal distribution, hence are possibly correlated, but it is not true otherwise in general, for instance if the rv are simply uncorrelated, not independent, but don't have a multivariate normal distribution.

Assuming that ε_lk[2] is ∝ CN(0, ...) then yes, this looks correct!

U have to check that your formulated distribution cdf must satisfies definition of cdf that is G(-inf)=0 and G(+ inf) =1, besides your cdf should be increasing.

Yes, but you need to specify clearly what is rho - pairwise correlation between consecutive error terms, and the first rho is zero, and no error at time zero.

Sorry. I don't agree with the other answers. A sum or a linear combination of normal random variables is not normal in general. It is normal if the random variables are independent or if they all belong to a (huge) multivariate normal distribution and, in the latter case, the random variables may be correlated.

Z=X+Y

if both X and Y are normal then Z is normally distributed with

Mean = Mean Y + Mean X

Var = Var Y + Var X + 2 Cov(X,Y)

The above notation look weird but in essence it is just an AR(1) with proper assumptions state clearly will do.

I missed out one point there is the person who raised the question needed to look into ~h[N]

Sorry Miang Hong Ngerng but this is wrong except if X and Y have together a bivariate normal distribution. Otherwise, the mean and the variance you stated are correct but not the normality.

this can be applied to any linear combination of normal variates by induction

Yes Miang Hong Ngerng if the variables have a multivariate normal distribution. Otherwise, you can only say that the mean is a linear combination of the means. For the variances, you have also a linear combination of the variances (with squared weights), provided the variables are uncorrelated since otherwise the covariances enter into the play. Normality of the linear combination is true if the variables are independent, since absence of correlation is not enough. The proofs are quite easy in each case. For normality, it comes from the fact that the multivariate density of independent variables is the product of the individual densities. Non-correlation is not enough.

Please review the attached file, please excuse my poor handwriting with ePen. Similar proof can be found in classical textbooks such as Mathematical Statistics Hogg, Craig and McKean.

Thank you Miang Hong Ngerng. No problem but your mistake is on line 7 of page 1. The sum of two correlated normal variables is not normal, in general. Except if the two variables are suppose to be distributed according to a bivariate normal distribution. See, e.g., https://planetmath.org/sumsofnormalrandomvariablesneednotbenormal and https://math.stackexchange.com/questions/4274029/sum-of-normal-random-variables-being-not-normal for a good counter-example.

You mention AR(1) models. The arguments above apply to forecast intervals. If the process is Gaussian (implying that every subset of the variables has a multivariate normal distribution), then you can obtain forecast intervals by using your arguments. Otherwise, you can write the mean and the variances of your forecasts but not obtain forecast intervals. I hope this is clearer.