Multivariate Cauchy estimator with scalar measurement and process noises

The conditional mean estimator for a n-state linear system with additive Cauchy measurement and process noises is developed. For the multi-variable system state, the characteristic function of the unnormalized conditional probability density function is sequentially propagated through measurement updates and dynamic state propagation, while expressing the resulting characteristic function in a closed analytical form. Continuity of this characteristic function and its first two derivatives at the origin of the spectral variable is proven. It is then used to determine the desired conditional mean and conditional variance in a closed analytical form to yield the sequential state estimator. A three-state dynamic system example demonstrates numerically the performance of the Cauchy estimator.