A note on defining positive definite functions

Abstract: A fundamental requirement in spatial statistics is that covariance functions are positive definite. While many positive definite functions are known for Euclidean spaces, their positive definiteness may not extend to non-Euclidean spaces. We present sufficient conditions to derive valid positive definite functions for spatial statistics on non-Euclidean geometries. Our approach leverages overlooked results due to Schoenberg (1938) to establish conditions under which covariance functions that are valid on Euclidean spaces remain valid on the domain of interest. This approach provides a more accessible and direct framework for spatial statisticians with diverse backgrounds.