Neighbourhoods of independence for random processes
The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced $α$-geometry, i.e., the $α$-curvature, $α$-Ricci curvature with its eigenvales and eigenvectors, the $α$-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere. We consider special cases as submanifolds and discuss their geometrical structures; one submanifold yields examples of neighbourhoods of the independent case for bivariate distributions having identical exponential marginals. Thus, since exponential distributions complement Poisson point processes, we obtain a means to discuss the neighbourhood of independence for random processes.