The representation of the uncertainty of a stochastic state by a Gaussian mixture is well-suited for nonlinear tracking problems in high dimensional data-starved environments such as space surveillance. In this paper, the framework for a Gaussian sum filter is developed emphasizing how the uncertainty can be propagated accurately over extended time periods in the absence of measurement updates. To achieve this objective, a series of metrics constructed from tensors of higher-order cumulants are proposed which assess the consistency of the uncertainty and provide a tool for implementing an adaptive Gaussian sum filter. Emphasis is also placed on the algorithm's potential for parallelization which is complemented by the use of higher-order unscented filters based on efficient multidimensional Gauss-Hermite quadrature schemes. The effectiveness of the proposed Gaussian sum filter is illustrated in a case study in space surveillance involving the tracking of an object in a six-dimensional state space.