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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: $Y_t = ε_{t} + ε_{t-1} + ...$

$= Covar(ε_{t-1},ε_{t-1}) + Covar(ε_{t-2},ε_{t-2}) + ...$

$= Var(ε_{t-1}) + Var(ε_{t-2}) + ...$

$= ∞ ???$

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Your $X_t$ seems to have infinite variance, let alone autocovariance. It is more usual to have $X_t = \phi X_{t-1} + ε_t$ where $|\phi | \lt 1$. –  Henry Mar 5 '12 at 8:32
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