Approach Theory Meets Probability

Abstract

Here we see that the construction of the weak* approach structure performed in the previous chapter, as in the classical topological setup, when restricted to probability measures allows for a quantification of the weak topology on probability measures. However we also consider other quantifications, depending on the problem at hand. In the first section we consider the general case of probability measures on a Polish space, in the second section we consider a quantification of convergence in probability for random variables. In the last section we prove an indexed version of the Lindeberg-Feller central limit theorem making use of a natural Lindeberg index indicating to what extent the Lindeberg condition is fulfilled.

The scientific imagination always restrains itself within the limits of probability.

(Thomas Huxley)

It is not certain that everything is uncertain.

(Blaise Pascal)