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On the null recurrence and transience of a first-order SETAR model

Published online by Cambridge University Press:  14 July 2016

Meihui Guo  and
Joseph D. Petruccelli
Meihui Guo
Affiliation:
Worcester Polytechnic Institute
Joseph D. Petruccelli*
Affiliation:
Worcester Polytechnic Institute
*
∗ Postal address: Mathematical Sciences Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA.
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Abstract

We consider the SETAR(l; 1, ···, 1) model: where – ∞= r0 < r1, < · ·· < rl = ∞ and for each j {ε t(j)} forms an i.i.d. zero-mean error sequence independent of {ε t(i)} for i ≠ j and having a density positive on the real line. Chan et al. (1985) obtained the region of the parameter space on which the process is ergodic, and showed the process to be transient on a subset of the remainder. They conjectured that the process was null recurrent everywhere else. In this paper we show that conjecture to be incorrect and under the assumption of finite variance of the error distributions we resolve the remaining questions of transience or null recurrence for this process.

Keywords

NON-LINEAR TIME SERIESERGODICITYMARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 584 - 592
Copyright
Copyright © Applied Probability Trust 1991 

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References

Chan, K. S., Petruccelli, J. D., Tong, H. and Woolford, S. W. (1985) A multiple threshold AR(1) model. J. Appl. Prob. 22, 267–279.Google Scholar
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Petruccelli, J. D. and Woolford, S. W. (1984) A threshold AR(1) model. J. Appl. Prob. 21, 270–286.Google Scholar
Tong, H. (1983) Threshold Models in Non-Linear Time Series Analysis. Lecture Notes in Statistics 21, Springer-Verlag, New York.Google Scholar
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