Long strange segments of a stochastic process

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 different rates in different 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.