  We present the first treatment of the arc length of the Gaussian Process (GP)
with more than a single output dimension. GPs are commonly used for tasks such
as trajectory modelling, where path length is a crucial quantity of interest.
Previously, only paths in one dimension have been considered, with no
theoretical consideration of higher dimensional problems. We fill the gap in
the existing literature by deriving the moments of the arc length for a
stationary GP with multiple output dimensions. A new method is used to derive
the mean of a one-dimensional GP over a finite interval, by considering the
distribution of the arc length integrand. This technique is used to derive an
approximate distribution over the arc length of a vector valued GP in
$\mathbb{R}^n$ by moment matching the distribution. Numerical simulations
confirm our theoretical derivations.
