In this paper we note that while the results of a 1981 paper of H. Tong's are generally valid and can be strengthened, there is a special case that behaves differently.

When considering the stability of a nonlinear time series, verifying aperiodicity, irreducibility and smoothness of the transitions for the corresponding Markov chain is often the first step. Here, we provide reasonably general conditions applicable to nonlinear autoregressive time series,...

With (X1, X2) in a stable domain of attraction and (Y1, Y2) independent of (X1, X2), conditions are given for which (X1Y1, X2Y2) is in the same domain and for which the same norming constants are applicable. For the case with no normal component, an alternative criterion for stable attraction...

We demonstrate a reliable and computationally feasible method for determining whether a given threshold autoregression autoregressive conditional heteroscedastic (AR-ARCH) model is ergodic, and for determining which moments exist when it is ergodic. This method may be used to delineate the...

In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictor which minimizes the dispersion (suitably defined) of the error distribution. When the linear process is driven by symmetric stable white noise this predictor minimizes the scale...

We present a formulation of subexponential and exponential tail behavior for multivariate distributions. The definitions are necessarily in terms of vague convergence of Radon measures rather than of ratios of distribution tails. With the proper setting, we show that if all one dimensional...