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  • 1.
    Aurzada, Frank
    et al.
    Technical University of Darmstadt, Germany .
    Dereich, Steffen
    University of Münster, Germany .
    Lifshits, Mikhail
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. St. Petersburg State University, Russia.
    Persistence probabilities for a Bridge of an integrated simple random walk2014In: Probability and Mathematical Statistics, ISSN 0208-4147, Vol. 34, no 1, p. 1-22Article in journal (Refereed)
    Abstract [en]

    We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability n(-1/2) to stay positive. This question is motivated by random polymer models and proves a conjecture by Caravenna and Deuschel.

  • 2.
    Liu, Zhenxia
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    ON THE LONGEST RUNS IN MARKOV CHAINS2018In: Probability and Mathematical Statistics, ISSN 0208-4147, Vol. 38, no 2, p. 407-428Article in journal (Refereed)
    Abstract [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 method 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 Gartner-Ellis theorem and the moment generating function.

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