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Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score

Author

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  • Chabriac, Claudie
  • Lagnoux, Agnès
  • Mercier, Sabine
  • Vallois, Pierre

Abstract

We calculate the density function of (U∗(t),θ∗(t)), where U∗(t) is the maximum over [0,g(t)] of a reflected Brownian motion U, where g(t) stands for the last zero of U before t, θ∗(t)=f∗(t)−g∗(t), f∗(t) is the hitting time of the level U∗(t), and g∗(t) is the left-hand point of the interval straddling f∗(t). We also calculate explicitly the marginal density functions of U∗(t) and θ∗(t). Let Un∗ and θn∗ be the analogs of U∗(t) and θ∗(t) respectively where the underlying process (Un) is the Lindley process, i.e. the difference between a centered real random walk and its minimum. We prove that (Un∗n,θn∗n) converges weakly to (U∗(1),θ∗(1)) as n→∞.

Suggested Citation

  • Chabriac, Claudie & Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2014. "Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4202-4223.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:12:p:4202-4223
    DOI: 10.1016/j.spa.2014.07.003
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    References listed on IDEAS

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    1. Daudin, Jean-Jacques & Etienne, Marie Pierre & Vallois, Pierre, 2003. "Asymptotic behavior of the local score of independent and identically distributed random sequences," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 1-28, September.
    2. M. P. Etienne & P. Vallois, 2004. "Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score," Methodology and Computing in Applied Probability, Springer, vol. 6(3), pages 255-275, September.
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    Cited by:

    1. Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2019. "Probability density function of the local score position," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3664-3689.
    2. Sabine Mercier & Grégory Nuel, 2022. "Duality Between the Local Score of One Sequence and Constrained Hidden Markov Model," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1411-1438, September.

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    1. Sabine Mercier & Grégory Nuel, 2022. "Duality Between the Local Score of One Sequence and Constrained Hidden Markov Model," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1411-1438, September.
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    3. M. P. Etienne & P. Vallois, 2004. "Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score," Methodology and Computing in Applied Probability, Springer, vol. 6(3), pages 255-275, September.
    4. Lagnoux, Agnès & Mercier, Sabine & Vallois, Pierre, 2019. "Probability density function of the local score position," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3664-3689.

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