Distribution of extremes and simulation of Gaussian processes. (English) Zbl 0940.60051

Seleznev, Oleg

Theory Probab. Math. Stat. 58, 161-169 (1999) and Teor. Jmovirn. Mat. Stat. 58, 149-157 (1998).

The author considers a Gaussian zero mean process \(X(t),t\in [0,T]\), with a covariance function \(R(t,s)\) and continuous sample paths. He studies the piecewise linear interpolation of this process and evaluation with a given accuracy of the distribution function of its maximum via Monte-Carlo method. Let \(F(a)\) be the distribution function of the random variable \(\eta=\max_{t\in[0,T]} X(t).\) Let \(X_{n}(t)\) be an approximation process \(X_{n}(t)= X_{n}(t,Y_1,\ldots,Y_{n})\), where \(Y_{i}=f\circ X\), \(i=1,\ldots,n\), is a set of functionals of an initial process, e.g. the process values in the interpolation knots \(Y_{i}=X(t_{i})\), \(i=1,\ldots,n.\) The approximation \(X_{n}\) is used to simulate the distribution \(\eta=\max_{t\in[0,T]}X(t)\) by simulating \(f\circ X_{n}.\) Let \(F_{n}(a)= P\{ f\circ X_{n}\leq a\}.\) The author describes an approach to estimate the distribution \(F(a)\) with an analysis of the approximation accuracy. The following assertion is the main result in this paper. If there exist \(L>0\), \(2\geq\alpha>0\), such that \[ E\bigl[(X^{(m)}(t)-X^{(m)}(s))^2\bigr]\leq L^2 |t-s|^{\alpha}, \tag{A} \] then there exists \(K=K(\alpha,a)>0\) such that \(|F(a)-F_{n}(a)|\leq K(\log n)^{1/2} n^{-(m+\alpha)/2)}\), and this rate of approximation is exact on the whole class of processes (A), \(m=0,1.\) The author presents several examples of an evaluation of distribution functions of the maximum of a continuous Gaussian process via simulation of sample paths.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G15	Gaussian processes
60G17	Sample path properties

Keywords:

simulation; Gaussian process; incremental variance function; sample paths; Monte-Carlo method