Estimates of distribution of the maximum of a Wiener two-parametric field given on a certain domain. (Russian) Zbl 0664.60054

Privorotskij, M. E.

Teor. Veroyatn. Mat. Stat., Kiev 38, 109-114 (1988).

Estimations of the distribution of the maximum of two-parameter Yeh- Wiener fields given on some domain are considered. It is shown that the probability of level crossing on the whole domain D is bounded by twice the probability of crossing the same level on the boundary of this domain or on the boundary without segments parallel to one of the axes. This inequality is applied to some concrete domains:
For the square: \(P\{\) max \(W\geq \lambda \}\leq 4\phi (\lambda)\), where \(\phi (\lambda)=(2\pi)^{-1/2}\int^{\infty}_{\lambda}\exp (-t^ 2/2)dt.\)
For the right-angled trapezium with altitude and base equal to one and sharp angle arc tg \(b\geq 45\circ:\) \(P\{\max W\geq \lambda \}\leq 2\phi (\lambda b^{1/2}(b-1)^{-1/2})+2\exp (-2\lambda^ 2b^{-1})\phi ((b- 2)\lambda (b(b-1))^{-1/2}),\)(here \(\phi (\infty)=0\) and \(\phi (- \infty)=1).\)
For the right-angled isosceles triangle: \(P\{\) max \(W\geq \lambda \}\leq 2\exp (-2\lambda^ 2).\)
For the convex quadrangle with two sharp angles arc tg \(a\geq 45\circ\) and arc tg \(b\geq 45\circ:\)
\[ P\{\max W\geq \lambda \}\leq 2\phi (\lambda (a+c)a^{-1}c^{- 1/2})+2\exp (-2\lambda^ 2a^{-1})\phi (\lambda (a-c)a^{-1}c^{- 1/2})+ \]
\[ 2\exp (-2\lambda^ 2b^{-1})\phi (\lambda (bc-1)b^{- 1}c^{-1/2})-2\exp (-2\lambda^ 2(a^{-1}+b^{-1}-2))\phi (-\lambda c^{-1/2}(b^{-1}-c-2));\quad c=(ab-a)(ab-1)^{-1}. \] For a fourth of a single circle a double inequality is obtained: \[ \phi (\lambda \sqrt{2})+2\exp (-2(\sqrt{2}-1)\lambda^ 2)\phi ((2-\sqrt{2})\lambda)- \exp (4\sqrt{2}(\sqrt{2}-1)\lambda^ 2)\phi (\lambda (4-\sqrt{2}))\leq \]
\[ P\{\max W\geq \lambda \}\leq 2\exp (-\lambda^ 2). \] If \(\lambda\) \(\uparrow \infty\), \(P\{\) max \(W\geq \lambda \}=f(\lambda)\exp (-\lambda^ 2)\), where \(f(\lambda)=O(1)\); \(f(\lambda)\neq o(\lambda^{-1})\).

Reviewer: M.E.Privorotskij

MSC:

60G60	Random fields
60E15	Inequalities; stochastic orderings
60J65	Brownian motion

Keywords:

two-parameter Yeh-Wiener fields; level crossing