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State-Discretization of V -Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution
Methodology and Computing in Applied Probability ( IF 0.9 ) Pub Date : 2019-11-04 , DOI: 10.1007/s11009-019-09746-0
Loic Hervé , James Ledoux

Let \((X_{n})_{n \in \mathbb {N}}\) be a V -geometrically ergodic Markov chain on a measurable space \(\mathbb {X}\) with invariant probability distribution π. In this paper, we propose a discretization scheme providing a computable sequence \((\widehat \pi _{k})_{k\ge 1}\) of probability measures which approximates π as k growths to infinity. The probability measure \(\widehat \pi _{k}\) is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of \((\widehat \pi _{k})_{k\ge 1}\) to π is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for π is known.

中文翻译:

V-遍历遍历马尔可夫链的状态离散和收敛到平稳分布

\((X_ {n})_ {n \ in \ mathbb {N}} \)是具有不变概率分布π的可测空间\(\ mathbb {X} \)上的V几何遍历马尔可夫链。在本文中,我们提出了一种离散化方案,该方案提供了概率度量的可计算序列\((\ widehat \ pi _ {k})_ {k \ ge 1} \),当k增长到无穷大时近似π。概率测度\(\ widehat \ pi _ {k} \)是根据有限马尔可夫链的不变概率分布计算得出的。\((\ widehat \ pi _ {k})_ {k \ ge 1} \)π的总变化的收敛速度给出。结果,研究了具有线性和非线性误差的一阶自回归过程的特殊情况。最后,提供了这种自回归过程的过程的图示,特别是在不知道π的明确公式的情况下。
更新日期:2019-11-04
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