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

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.

### MSC:

 60G15 Gaussian processes 60G17 Sample path properties