Extremes of autoregressive threshold processes

In this paper we study the tail and the extremal behavior of stationary solutions of autoregressive threshold (TAR) models. It is shown that a regularly varying noise sequence leads in general only to an O-regularly varying tail of the stationary solu- tion. Under further conditions on the partition, it is however shown that TAR(S, 1) models of order 1 with S regimes have regularly varying tails, provided the noise se- quence is regularly varying. In these cases, the finite dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in L(γ) with γ > 0. In that case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.     «

In this paper we study the tail and the extremal behavior of stationary solutions of autoregressive threshold (TAR) models. It is shown that a regularly varying noise sequence leads in general only to an O-regularly varying tail of the stationary solu- tion. Under further conditions on the partition, it is however shown that TAR(S, 1) models of order 1 with S regimes have regularly varying tails, provided the noise se- quence is regularly varying. In these cases, the finite dimensional dist...     »