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  • 1.
    Gao, Fuchang
    et al.
    University of Idaho.
    Liu, Zhenxia
    Blåeldsvägen 12B, Sturefors, Sweden.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Conditional persistence of Gaussian random walks2014In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 19, no 70, p. 1-9Article in journal (Refereed)
    Abstract [en]

    Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. standard Gaussian random variables, let $S_n=\sum_{i=1}^nX_i$ be the Gaussian random walk, and let $T_n=\sum_{i=1}^nS_i$ be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence:\begin{align*}\mathbb{P}\left\{\max_{1\leq k \leq n}T_{k} \leq 0\,\,\Big|\,\,T_n=0,S_n=0\right\}&\lesssim n^{-1/2},\\\mathbb{P}\left\{\max_{1\leq k \leq 2n}T_{k} \leq 0\,\,\Big|\,\,T_{2n}=0,S_{2n}=0\right\}&\gtrsim\frac{n^{-1/2}}{\log n},\end{align*}for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008).

  • 2.
    Gao, FuChang
    et al.
    Univ Idaho, Dept Math, Moscow, ID 83844 USA.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Upper tail probabilities of integrated Brownian motions2015In: Science China Mathematics, ISSN 1674-7283, Vol. 58, no 5, p. 1091-1100Article in journal (Refereed)
    Abstract [en]

    We obtain new upper tail probabilities of m-times integrated Brownian motions under the uniform norm and the L (p) norm. For the uniform norm, Talagrands approach is used, while for the L (p) norm, Zolotares approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variables in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of m-times integrated Brownian motions are presented as well.

  • 3.
    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.
    A general large deviation principle for longest runs2016In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 110, p. 128-132Article in journal (Refereed)
    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.

  • 4.
    Liu, Zhenxia
    et al.
    Blåeldsvägen 12B, Sturefors, Sweden.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Probabilities of hitting a convex hull2014In: Comptes rendus. Mathematique, ISSN 1631-073X, E-ISSN 1778-3569, Vol. 352, no 11, p. 935-940Article in journal (Refereed)
    Abstract [en]

    In this note, we consider the non-negative least-square method with a random matrix. This problem has connections with the probability that the origin is not in the convex hull of many random points. As related problems, suitable estimates are obtained as well on the probability that a small ball does not hit the convex hull.

  • 5.
    Privault, Nicolas
    et al.
    Nanyang Technological University.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Zambrini, Jean-Claude
    Universidade de Lisboa.
    Feynman-Kac formula for Levy processesand semiclassical (Euclidean) momentum representation2014In: Markov Processes and Related Fields, ISSN 1024-2953, Vol. 20, no 3, p. 577-600Article in journal (Refereed)
    Abstract [en]

    We prove a version of the Feynman-Kac formula for Levy processes andintegro-differential operators, with application to the momentum representationof suitable quantum (Euclidean) systems whose Hamiltonians involve L´evytypepotentials. Large deviation techniques are used to obtain the limitingbehavior of the systems as the Planck constant approaches zero. It turns outthat the limiting behavior coincides with fresh aspects of the semiclassical limitof (Euclidean) quantum mechanics. Non-trivial examples of Levy processes areconsidered as illustrations and precise asymptotics are given for the terms inboth configuration and momentum representations.

  • 6.
    Privault, Nicolas
    et al.
    School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Zambrini, Jean-Claude
    Group of Mathematical Physics, University of Lisbon, Lisbon, Portugal.
    Large deviations for Bernstein bridges2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 5, p. 1285-1305Article in journal (Refereed)
    Abstract [en]

    Bernstein processes over a finite time interval are simultaneously forward and backward Markov processes with arbitrarily fixed initial and terminal probability distributions. In this paper, a large deviation principle is proved for a family of Bernstein processes (depending on a small parameter ħ which is called the Planck constant) arising naturally in Euclidean quantum physics. The method consists in nontrivial Girsanov transformations of integral forms, suitable equivalence forms for large deviations and the (local and global) estimates on the parabolic kernel of the Schrödinger operator.

  • 7.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Exact upper tail probabilities of random series2015In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 99, p. 13-19Article in journal (Refereed)
    Abstract [en]

    In this paper, we obtain new estimates on upper tail probabilities of suitable random series involving a distribution having an exponential tail. These estimates are exact, and the distribution is not heavy-tailed.

  • 8.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
    Large deviations for Markov bridges with jumps2014In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 416, no 1, p. 1-12Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider a family of Markov bridges with jumps constructed from truncated stable processes. These Markov bridges depend on a small parameter h greater than 0, and have fixed initial and terminal positions. We propose a new method to prove a large deviation principle for this family of bridges based on compact level sets, change of measures, duality and various global and local estimates of transition densities for truncated stable processes.

  • 9.
    Yang, Xiangfeng
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    On the large deviation principle of generalized Brownian bridges2015In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 430, no 2, p. 845-856Article in journal (Refereed)
    Abstract [en]

    In this paper we consider a family of generalized Brownian bridges with a small noise, which was used by Brennan and Schwartz [3] to model the arbitrage profit in stock index futures in the absence of transaction costs. More precisely, we study the large deviation principle of these generalized Brownian bridges as the noise becomes infinitesimal. (C) 2015 Elsevier Inc. All rights reserved.

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