  Many applications require stochastic processes specified on two- or
higher-dimensional domains; spatial or spatial-temporal modelling, for example.
In these applications it is attractive, for conceptual simplicity and
computational tractability, to propose a covariance function that is separable;
e.g., the product of a covariance function in space and one in time. This paper
presents a representation theorem for such a proposal, and shows that all
processes with continuous separable covariance functions are second-order
identical to the product of second-order uncorrelated processes. It discusses
the implications of separable or nearly separable prior covariances for the
statistical emulation of complicated functions such as computer codes, and
critically reexamines the conventional wisdom concerning emulator structure,
and size of design.
