Comment by cirpis

Two things: 1. The result is asymptotical, i.e. holds as number of samples approach infinity.

2. The result is an "almost surely" result, i.e. in the collection of all possible infinite samples, the set of samples for which it fails has 0 measure. In non technical terms this means that it works for typical random samples and may not work for handpicked counterexamples.

In our particular case let f=Sha256. Then X must be discrete, i.e. a natural number. Now the particulars depend on the distribution on X, but the general idea is that since we have discrete values, the probability that we get an infinite sample where the values tend to infinity is 0. So we get that in a typical sample theres going to be an infinitude of x ties and furthermore most x values arent too large (in a way you can make precise), so the tie factors l_i dominate since there just arent that many distinct values encountered total. And so we get that the coefficient tends to 1.