Large sample theory for statistics of stable moving averages

Abstract

We study the limit behavior of the partial sums, sample variance, and periodogram of the stable moving average process x(t)= ∫ ψ(t + x)double struck M sign (dx) 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" α, which is an unknown parameter of the model. Through the employment of self-normalization, this number α 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 α. New techniques, which are easily generalizable to a random field model, are presented to prove these results.