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  • 1.
    Liu, Zhenxia
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Mbokoma, Mainza
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    An improvement on the large deviations for longest runs in Markov chains2023Ingår i: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 193, artikel-id 109737Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Large deviations for longest success runs L(n) in Markov chains have been previously studied in Liu and Yang (2018) and Liu and Zhu (2020) under a technical assumption p10 < p00 +p11, with pij denoting the transition probability from i to j. In this note, we prove that all the results in Liu and Yang (2018) and Liu and Zhu (2020) still hold even without such an assumption. The main step in the proof is to derive an improved global estimation for the distribution function of L(n) without this assumption, which might be of independent interest.

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  • 2.
    Liu, Zhenxia
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Optimal Monte Carlo method in estimating areas2021Ingår i: Results in Applied Mathematics, ISSN 2590-0374, Vol. 12, artikel-id 100205Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    It is well known that Monte Carlo method can be used to estimate the area of a region which cannot be computed directly. There are a lot of ways to choose a larger region whose area is computable when one performs Monte Carlo method, but which region is the best? In this note, we find a best region in terms of fastest speed of convergence in probability, with the help of large deviations.

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  • 3.
    Liu, Zhenxia
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Zhu, Yurong
    Beihang Univ, Peoples R China.
    Large deviations for longest runs in Markov chains2020Ingår i: JOURNAL OF APPLIED ANALYSIS, ISSN 1425-6908, Vol. 26, nr 2, s. 309-314Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 (2016), 128-132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 (2018), no. 2, 407-428].

  • 4.
    Liu, Zhenxia
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    ON THE LONGEST RUNS IN MARKOV CHAINS2018Ingår i: Probability and Mathematical Statistics, ISSN 0208-4147, Vol. 38, nr 2, s. 407-428Artikel i tidskrift (Refereegranskat)
    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.

  • 5.
    Liu, Zhenxia
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Wang, Zhi
    Ningbo University of Technology, Peoples R China.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    A Gaussian expectation product inequality2017Ingår i: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 124Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let (X-1,. . ., X-n) be any n-dimensional centered Gaussian random vector, in this note the following expectation product inequality is proved: E Pi (n)(j=1) f(j)( X-j) amp;gt;= Pi (n)(j=1) Ef(j)(X-j) for functionsh, 1 amp;lt;= j amp;lt;= n, taking the forms f(j)(x) = integral(infinity)(0) where mu(j), 1 amp;lt;= j amp;lt;= n, are finite positive measures. The motivation of studying such an inequality comes from the Gaussian correlation conjecture (which was recently proved) and the Gaussian moment product conjecture (which is still unsolved). Several explicit examples of such functions f(j) are given. The proof is built on characteristic functions of Gaussian random variables. (C) 2017 Elsevier B.V. All rights reserved.

  • 6.
    Liu, Zhenxia
    et al.
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    A general large deviation principle for longest runs2016Ingår i: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 110, s. 128-132Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this note we prove a general large deviation principle (LDP) for the longest success run in a sequence of independent Bernoulli trails. This study not only recovers several recently derived LDPs, but also gives new LDPs for the longest success run. The method is based on the Bryc’s inverse Varadhan lemma, which can be intuitively generalized to the longest success run in a two-state (success and failure) Markov chain.

  • 7.
    Konstantopoulos, Takis
    et al.
    Uppsala University, Sweden.
    Liu, Zhenxia
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    Yang, Xiangfeng
    Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
    LAPLACE TRANSFORM ASYMPTOTICS AND LARGE DEVIATION PRINCIPLES FOR LONGEST SUCCESS RUNS IN BERNOULLI TRIALS2016Ingår i: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 53, nr 3, s. 747-764Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log(1/p) n and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

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