Approximation formulas for the moments of the boundary functional of a Gaussian random walk with positive drift by using Siegmund’s formula. (English) Zbl 07551980
Summary: In this study, a boundary functional \((SN(x))\) are mathematically constructed for a Gaussian random walk (GRW) with positive drift \(\beta\) and first four moments of the functional \(SN(x)\) are expressed in terms of ladder variables based on Dynkin Principle. Moreover, approximation formulas for first three moments of ladder height \(\chi_1^+\) are proposed based on the formulas of D. Siegmund [Adv. Appl. Probab. 11, 701–719 (1979; Zbl 0422.60053)] when \(\beta \downarrow 0\). Finally, approximation formulas for the first four moments of the boundary functional \(SN(x)\) are obtained by using Siegmund formulas and meta modeling, when \(\beta \in [0.1, 3.6]\).
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60K15 | Markov renewal processes, semi-Markov processes |
| 60F99 | Limit theorems in probability theory |
Keywords:
boundary functional; Dynkin principle; Gaussian random walk; ladder height; meta-modeling; positive drift; Reimann zeta-functionCitations:
Zbl 0422.60053References:
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