Abstract. Sparse Gaussian processes are generally constructed using finite-dimensional marginal distributions, but this is not the only way to think about them. Pathwise conditioning enables one to understand these approximations using the perspective of random functions. We use this perspective to study the numerical stability of scalable sparse approximations. We derive sufficient and in certain cases necessary conditions on the inducing points for the computations performed to be numerically stable. We propose an automated method for computing inducing points satisfying these conditions. Our results show that, in geospatial settings, sparse approximations with guaranteed numerical stability often perform comparably to those without.