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February, 1995 An Error Analysis for the Numerical Calculation of Certain Random Integrals: Part 1
Loren D. Pitt, Raina Robeva, Dao Yi Wang
Ann. Appl. Probab. 5(1): 171-197 (February, 1995). DOI: 10.1214/aoap/1177004835


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.


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Loren D. Pitt. Raina Robeva. Dao Yi Wang. "An Error Analysis for the Numerical Calculation of Certain Random Integrals: Part 1." Ann. Appl. Probab. 5 (1) 171 - 197, February, 1995.


Published: February, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0821.62058
MathSciNet: MR1325048
Digital Object Identifier: 10.1214/aoap/1177004835

Primary: 62M40
Secondary: 62M20

Keywords: numerical integration , optimal prediction of random fields and integrals , spatial statistics

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.5 • No. 1 • February, 1995
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