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A Note on Geometric Ergodicity of Markov Chains

盧炤傑(Chao-Chieh Lu)

指導教授 ： 許順吉

國立中央大學/理學院/數學系/碩士(2014年)

摘要

對於可數多個態、同質的馬可夫鏈我們已經有一些基本的認知，而且由D. G. Kendall 證明一個對於數列 (p_ij^((n) )-π_ij) 幾何收斂的‘solidarty theorem’。我們想檢驗幾何遍地性以及去得到馬可夫鏈的幾何收斂參數 ρ_ij。因此，我們在中間建構並且推廣一些的馬可夫鏈的極限定理；此外，我們可以在一個共同的圓 C_(R^' ) (R^'>R) 使生成函數P_00 (z)延拓成亞純函數(meromorphic function)使其在 z=R 有一個簡單極(simple pole)。最後，我們去推論出幾何遍地性以及幾何收斂參數 ρ_ij。

關鍵字

馬可夫鏈 ； 幾何遍地性 ； 收斂參數

並列摘要

We already had known about some basic understanding of homogeneous Markov chain with countable state space, and D. G. Kendall has proved a 'solidarity theorem' for geometric convergence of the sequences (p_ij^((n) )-π_ij ) with convergence parameter ρ_ij. We shall investigate the geometric ergodicity and the convergence parameters ρ_ij. Therefore, we construct and generate some theorems of Markov chain. Also, we extend the genereating function P_00 (z) as a meromorphic function within a common disk C_(R^' ) (R^'>R) which it has only simple pole at z=R. Finally, we deduce some results for geometric ergodicity and convergence parameters ρ_ij.

並列關鍵字

Markov Chain ； Geometric Ergodicity ； Convergence parameter

參考文獻

[1] D. Vere-Jones, Geometric ergodicity in denumerable Markov chains. Quarterly Journal of Mathematics (Oxford, Series 2, 1960), 13, 7-28.

[2] K. L. Chung, Markov Chains with Stationary Transition Probabilities (Berlin:1960)

[3] G. H. Hardy, Divergent Series (Oxford, 1949).

[4] C. Derman, 'A solution to a set of fundamental equations in Markov chains', Proc. American Math. Soc. 5 (1954) 332-4.

[5] D. G. Kendall, 'Unitary dilations of Markov transition operators and the corresponding integral representations for transition-probability matrices', in U. Grenander (ed.), Probability and statistics (Stockholm: Almqvist and Wiksell; New York, 1959).