Long Strange Segments of a Stochastic Process and Long Range Dependence

We study long strange intervals in a linear stationary stochastic process with regularly varying tails. It turns out that the length of the longest strange interval grows, as a function of the sample size, at diierent rates in diierent parts of the parameter space. We argue that this phenomenon may be viewed in a fruitful way as a phase transition between short and long range dependence. We prove a limit theorem that may form a basis for statistical detection of long range dependence.