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The Weak Convergence Theorem for the Distribution of the Maximum of a Gaussian Random Walk and Approximation Formulas for its Moments

Author

Listed:
  • Fikri Gökpınar

    (Gazi University)

  • Tahir Khaniyev

    (TOBB University of Economics and Technology
    Azerbaijan National Academy of Sciences)

  • Zulfiyya Mammadova

    (Karadeniz Technical University)

Abstract

In this study, asymptotic expansions of the moments of the maximum (M(β)) of Gaussian random walk with negative drift ( − β), β > 0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(β) ≡ 2 β M(β) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter β ∈ (0.5, 3.2] using meta-modeling.

Suggested Citation

  • Fikri Gökpınar & Tahir Khaniyev & Zulfiyya Mammadova, 2013. "The Weak Convergence Theorem for the Distribution of the Maximum of a Gaussian Random Walk and Approximation Formulas for its Moments," Methodology and Computing in Applied Probability, Springer, vol. 15(2), pages 333-347, June.
    • Handle: RePEc:spr:metcap:v:15:y:2013:i:2:d:10.1007_s11009-011-9240-0
    DOI: 10.1007/s11009-011-9240-0
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