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

Research paper by Fikri Gökpınar, Tahir Khaniyev, Zulfiyya Mammadova

Indexed on: 13 Jul '11Published on: 13 Jul '11Published in: Methodology and computing in applied probability

#### 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.