We study the limit behavior of the partial sums, sample variance, and periodogram of the stable moving average process   explored in Resnick, S., Samorodnitsky, G., and Xue, F. (1999). How misleading can sample ACF's of stable MA's be? (Very!). Annals of Applied Probability, 9(3), 797-817. Each of these statistics has a rate of convergence involving the “characteristic exponent” , which is an unknown parameter of the model. Through the employment of self-normalization, this number can be removed from the rate of convergence; the various limit distributions can then be approximated via subsampling. As a result, statistical inference for the mean can be conducted without knowledge (or explicit estimation) of . New techniques, which are easily generalizable to a random field model, are presented to prove these results.