Information geometry and asymptotic geodesics on the space of normal distributions

The family $$\mathcal {N}$$ of n-variate normal distributions is parameterized by the cone positive definite symmetric $$n\times n$$ -matrices and n-dimensional real vector space. Equipped with Fisher information metric, becomes a Riemannian manifold. As such, it diffeomorphic, but not isometric, to space $$\mathrm {Pos}_1(n+1,\mathbb {R})$$ unimodular $$(n+1)\times (n+1)$$ -matrices. computation distances in metric for $$n>1$$ presents some difficulties, Lovrič et al. (J Multivar Anal 74:36–48, 2000) proposed use Killing on as an alternative which are easier compute. In this work, we survey geometric properties provide quantitative analysis defect certain geodesics be metric. We find that these approximation indeed justified long distances.