When calculating the lag 1 autocovariance of a simple AR process, if I define my $X_{t}$'s in terms of $ε_t$'s, I get something that looks a lot like an MA process with an infinite sum of standard normal variables with an infinite variance. Where am I going wrong with this?
$X_t = X_{t-1} + ε_t$
$Covar(X_t, X_{t-1}) $
$= Covar(X_{t-1} + ε_t,X_{t-2} + ε_{t-1})$
$= Covar(X_{t-2} + ε_{t-1} + ε_t, X_{t-3}+ ε_{t-2} + ε_{t-1})$
This is starting to look like an MA process: $Y_t = ε_{t} + ε_{t-1} + ...$
$= Covar(ε_{t-1},ε_{t-1}) + Covar(ε_{t-2},ε_{t-2}) + ...$
$= Var(ε_{t-1}) + Var(ε_{t-2}) + ...$
$= ∞ ???$