Probability estimates of exceeding a level by Gaussian random series. (Russian) Zbl 0696.60040

Pashko, A. A.

Teor. Veroyatn. Mat. Stat., Kiev 41, 80-88 (1989).

Let \(\xi (t)=\sum^{\infty}_{k=1}f_ k(t)\xi_ k\), \(t\in R^ 1\), be a Gaussian process with E \(\xi\) (t)\(=0\), and \(\{f_ k(t)\}_{k\geq 1}\) be a complete orthonormal sequence of functions from some class B. Let \[ \xi_ N(t)=\sum^{N}_{k=1}f_ k(t)\xi_ k,\quad t\in R^ 1, \] be a Monte-Carlo model for a process \(\xi\) (t), where N is such that \[ P\{\| \sum^{\infty}_{k=N+1}f_ k(t)\xi_ k\|_ C\geq x_ p\}<p \] for some p and \(x_ p\). The exponential bounds for \[ P\{\| c(t)\sum^{n}_{k=m}f_ k(t)\xi_ k\|_ C\geq x\}, \] where \(| c(t)| \leq 1\), \(\int^{\infty}_{-\infty}| c(t)| dt<\infty\), are obtained.

Reviewer: N.Leonenko

MSC:

60G15	Gaussian processes
60F10	Large deviations
65C05	Monte Carlo methods

Keywords:

Gaussian process; Monte-Carlo model