This paper establishes a necessary and sufficient condition for geometrical ergodicity for the general first-order threshold autoregressive processes. This is achieved by investigating the nonlinear dynamic behavior generated by the delay parameter of a threshold model. The ergodic region turns out to be unbounded which is different from that of a linear process.
"On the Ergodicity of Tar(1) Processes." Ann. Appl. Probab. 1 (4) 613 - 634, November, 1991. https://doi.org/10.1214/aoap/1177005841