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Asymptotic behavior of the local score of independent and identically distributed random sequences

Author

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  • Daudin, Jean-Jacques
  • Etienne, Marie Pierre
  • Vallois, Pierre

Abstract

Let (Xn)n[greater-or-equal, slanted]1 be a sequence of real random variables. The local score is Hn=max1[less-than-or-equals, slant]i +[infinity], where B1*=max0[less-than-or-equals, slant]u[less-than-or-equals, slant]1 Bu and (Bu,u[greater-or-equal, slanted]0) is a standard Brownian motion, B0=0. If (Xn)n[greater-or-equal, slanted]1 a sequence of i.i.d. random variables, and Var(X1)=[sigma]2>0, we prove the convergence of to [sigma][xi][delta]/[sigma] where [xi][gamma]=max0[less-than-or-equals, slant]u[less-than-or-equals, slant]1 {(B(u)+[gamma]u)-min0[less-than-or-equals, slant]s[less-than-or-equals, slant]u(B(s)+[gamma]s)}. We approximate the probability distribution function of [xi][gamma] and we determine the asymptotic behavior of P([xi][gamma][greater-or-equal, slanted]a), a-->+[infinity].

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:107:y:2003:i:1:p:1-28
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    Citations

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    Cited by:

    1. McGill, Paul, 2006. "An asymptotic estimate for Brownian motion with drift," Statistics & Probability Letters, Elsevier, vol. 76(11), pages 1164-1169, June.
    2. 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.
    3. 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.
    4. 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.
    5. 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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