Tail index estimation for dependent data

A popular estimator of the index of regular variation in heavy-tailed model s is Hill's estimator. We discuss the consistency of estimator when it is a pplied to certain classes of heavy-tailed stationary processes. One class o f processes discussed consists of processes which can be appropriately appr oximated by sequences of m-dependent, random variables and special cases of our results show the consistency of Hill's estimator for (i) infinite movi ng averages with heavy-tail innovations, (ii) a simple stationary bilinear model driven by heavy-tail noise variables and (iii) solutions of stochasti c difference equations of the form Y-t = A(t)Y(t-1) + Z(t), -infinity < t < infinity where {(A(n), Z(n)), -infinity < n < infinity} are lid and the Z's have reg ularly varying tail probabilities. Another class of problems where our meth ods work successfully are solutions of stochastic difference equations such as the ARCH process where the process cannot be successfully approximated by m-dependent random variables. A final class of models where Hill estimat or consistency is proven by our tail empirical process methods is the class of hidden semi-Markov models.