You may use the line which is seen right above (before) the section "Correlated random variables" in the following link
https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables#Correlated_random_variables
Thus, Z ~ N(0, a+b), given that X and Y are independent.
Steftcho P. Dokov actually the problem I was facing was a reparameterization trick. So, Z(t) = Z(t-1) + sqrt(a)ε(t-1) + sqrt(b) ε(t-2);
where ε~N(0, I). It is used in diffusion models.
Well, your second equation, Z(t) = Z(t-1) + sqrt(a)ε(t-1) + sqrt(b) ε(t-2), looks like something similar to GARCH(1,2) or GARCH(2,1) model. Anyway, eventually, first, you can argue that the difference Z(t) - Z(t-1) is a normal random variable provided ε's are normal. And, second, applaying / using the Cramér's decomposition theorem
https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_decomposition_theorem
yields that Z(t) and Z(t-1) are themselves normal - - hopefully, this helps somewhat.
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