On the convergence rate of ordinal comparisons of random variables

The asymptotic exponential convergence rate of ordinal comparisons follows from well-known results in large deviations theory, where the critical condition is the existence of a finite moment generating function. In this note, we show that this is both a necessary and sufficient condition, and also show how one can recover the exponential convergence rate in cases where the moment generating function is not finite. In particular, by working with appropriately truncated versions of the original random variables, the exponential convergence rate can be recovered.