The Annals of Applied Probability

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

Loren D. Pitt, Raina Robeva, and Dao Yi Wang

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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

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

First available in Project Euclid: 19 April 2007

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Primary: 62M40: Random fields; image analysis
Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Spatial statistics numerical integration optimal prediction of random fields and integrals


Pitt, Loren D.; Robeva, Raina; Wang, Dao Yi. An Error Analysis for the Numerical Calculation of Certain Random Integrals: Part 1. Ann. Appl. Probab. 5 (1995), no. 1, 171--197. doi:10.1214/aoap/1177004835.

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