How misleading can sample acf's of stable MAs be? (very!)

For the stable moving average process X-t = integral(-infinity)(infinity) f (t + x) M(dx), t= 1,2,...., we find the weak limit of its sample autocorrel ation function as the sample size n increases to infinity. It turns out tha t, 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.