On the longest runs in Markov chains

• Autorzy: Liu Z. , Yang X.

Identyfikatory

DOI

10.19195/0208-4147.38.2.8

• Języki publikacji: EN

Abstrakty

EN

In the first n steps of a two-state (success and failure) Markov chain, the longest success run L(n) has been attracting considerable attention due to its various applications. In this paper, we study L(n) in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our metod on the moment generating function is based on a global estimate of the cumulative distribution function of L(n) proposed in this paper, and the proofs of the large deviations include the Gärtner-Ellis theorem and the moment generating function.

Słowa kluczowe

EN

longest run; moment generating function; large deviation principle; Markov chain

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 2
• Strony: 407--428
• Opis fizyczny: Bibliogr. 14 poz., tab.

Twórcy

autor

Liu Z.

• Department of Mathematics, Linköping University, 581 83 Linköping, Sweden

autor

Yang X.

• Department of Mathematics, Linköping University, 581 83 Linköping, Sweden

Bibliografia

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• [8] L. Holst and T. Konstantopoulos, Runs in coin tossing: A general approach for deriving distributions for functionals, J. Appl. Probab. 52 (3) (2015), pp. 752-770.
• [9] T. Konstantopoulos, Z. Liu, and X. Yang, Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials, J. Appl. Probab. 53 (3) (2016), pp. 747-764.
• [10] Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 (2016), pp. 128-132.
• [11] Y. Mao, F. Wang, and X. Wu, Large deviation behavior for the longest head run in an IID Bernoulli sequence, J. Theoret. Probab. 28 (1) (2015), pp. 259-268.
• [12] A. Rényi, Foundations of Probability, Holden-Day, San Francisco 1970.
• [13] R. E. Young and A. L. Sweet, Confidence intervals for transition probabilities in two-state Markov chains. I, Comm. Statist. A-Theory Methods 11 (2) (1982), pp. 165-179.
• [14] Y. Z. Zhang and X. Y. Wu, Some results associated with the longest run in a strongly ergodic Markov chain, Acta Math. Sin. (Engl. Ser.) 29 (10) (2013), pp. 1939-1948.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).