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Correction to: Random Coefficient Autoregressive Processes: a Markov Chain Analysis of Stationarity and Finiteness of Moments by Paul D. Feigin and Richard L. Tweedie J. Time Series Anal., Vol. 6, No. 1 (1985)
Journal of Time Series Analysis ( IF 0.9 ) Pub Date : 2020-03-16 , DOI: 10.1111/jtsa.12521
Paul D. Feigin 1
Affiliation  

The statement of Theorem 1 in Feigin and Tweedie (1985) is incomplete if it is to be a consequence of Theorem 4(ii) in Tweedie (1983a). Theorem 1 should read as below with (2.1a) and aperiodicity being the extra conditions. Indeed, we provide a counterexample to the original formulation.

Theorem 1.Suppose that {Xt} is an aperiodic Feller chain, that there exists a measure φ and a compact set A with φ(A)>0 such that

  • (i)

    {Xt} is φ‐irreducible

  • (ii)

    there exists a non‐negative continuous function g : E R satisfying

    g ( x ) 1 for x A (2.1)
    E { g ( X t ) | X t 1 = x } < for x A (2.1a)

and for some δ>0

E { g ( X t ) | X t 1 = x } ( 1 δ ) g ( x ) for x A c . (2.2)
Then {Xt} is geometrically ergodic.



中文翻译:

校正至:随机系数自回归过程:平稳性和矩有限度的马尔可夫链分析,作者:Paul D. Feigin和Richard L. Tweedie J.时间序列分析,第一卷。6,No.1(1985)

如果是Tweedie(1983a)中定理4(ii)的结果,则Feigin和Tweedie(1985)中的定理1的陈述是不完整的。定理1应读为(2.1a),且非周期性为附加条件。确实,我们提供了原始公式的反例。

定理1.假设{ X t }是一个非周期Feller链,存在一个测度φ和一个紧集A,φA)> 0使得

  • (一世)

    { X t }是φ-不可约的

  • (ii)

    存在一个非负连续函数 G Ë [R 满意的

    G X 1个 对于 X 一种 (2.1)
    Ë { G X Ť | X Ť - 1个 = X } < 对于 X 一种 (2.1a)

并且对于δ > 0

Ë { G X Ť | X Ť - 1个 = X } 1个 - δ G X 对于 X 一种 C (2.2)
那么{ X t }是几何遍历的。

更新日期:2020-03-16
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