How misleading can sample ACFs of stable MAs be? (Very!)

For the stable moving average process $$X^t = \int_{-\infty}^{\infty} f(t + x)M(dx), t = 1, 2,\dots,$$ we find the weak limit of its sample autocorrelation function as the sample size n increases to $\infty$. It turns out that, as a rule, this limit is random! This shows how dangerous it is to rely on sample correlation as a model fitting tool in the heavy tailed case. We discuss for what functions f this limit is nonrandom for all (or only some--this can be the case, too!) lags.