## The Annals of Applied Probability

### An Error Analysis for the Numerical Calculation of Certain Random Integrals: Part 1

#### Abstract

For a wide sense stationary random field $\Phi = \{\phi(x): x \in R^2\}$, we investigate the asymptotic errors made in the numerical integration of line integrals of the form $\int_\Gamma f(x)\phi(x)d\sigma(x)$. It is shown, for example, that if $f$ and $\Gamma$ are smooth, and if the spectral density $\rho(\lambda)$ satisfies $\rho(\lambda)\approx k|\lambda|^{-4}$ as $\lambda \rightarrow \infty$, then there is a constant $c'$ with $N^3E|\int_\Gamma f(x)\phi(x)d\sigma(x) - \sum \beta_{j\varphi}(x_j)|^2 \geq c'N^{-3}$ for all finite sets $\{x_j: 1 \leq j\leq N\}$ and all choices of coefficients $\{\beta_j\}$. And, if any fixed parameterization $x(t)$ of $\Gamma$ is given and the integral $\int^1_0 f(x(t))\phi(x(t))|x'(t)|dt$ is numerically integrated using the midpoint method, the exact asymptotics of the mean squared error is derived. This leads to asymptotically optimal designs, and generalizes to other power laws and to nonstationary and nonisotropic fields.

#### Article information

Source
Ann. Appl. Probab., Volume 5, Number 1 (1995), 171-197.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177004835

Digital Object Identifier
doi:10.1214/aoap/1177004835

Mathematical Reviews number (MathSciNet)
MR1325048

Zentralblatt MATH identifier
0821.62058

JSTOR