Limit theory for martingale transforms with heavy-tailed noise

A limit theorem for partial sums of martingale transforms with multiplicative noise and ergodic transform processes is established, which leads to regularly varying rates and stable limits. Its establishment is facilitated by the derivation of an extension to empirical distributions of Breiman’s Theorem along with the Principle of Conditioning. The theorem is applied for the development of the limit theory of the least squares estimator in regressions with heavy tailed noise, as well as of the limit theory of the Gaussian QMLE in GARCH-type models. Depending on the index of stability, regularly varying rates and asymptotic stable distributions, or inconsistency, are obtained.