Mean and variance of the arc length of a Gaussian process on a finite interval†

Abstract

The first and second moments of the distribution function of the arc length of a Gaussian process on a finite interval are obtained in terms of the covariance function of the derivative process. A closed expression (in terms of a modified Bessel function) was obtained for the first moment; however, the second moment had to be evaluated numerically. Numerical calculations were carried out for three typical covariance functions.

Notes

† Communicated by the Authors.; ‡ On leave of absence from Itek Corporation, Lexington, Mass.