Best predictors in logarithmic distance between positive random variables

The metric properties of the set in which random variables take their values lead to relevant probabilistic concepts. For example, the mean of a random variable is a best predictor in that it minimizes the L₂ distance between a point and a random variable. Similarly, the median is the same concept but when the distance is measured by the L₁ norm.

Also, a geodesic distance can be defined on the cone of strictly positive vectors in ℝⁿ in such a way that, the minimizer of the distance between a point and a collection of points is their geometric mean.

That geodesic distance induces a distance on the class of strictly positive random variables, which in turn leads to an interesting notions of conditional expectation (or best predictors) and their estimators. It also leads to different versions of the Law of Large Numbers and the Central Limit Theorem. For example, the lognormal variables appear as the analogue of the Gaussian variables for version of the Central Limit Theorem in the logarithmic distance.