Not All Sample Statistics
Approximate a Normal Distribution
as Sample Size Increases

Consider the distribution of sample range, the difference between the sample maximum and the sample minimum, for samples of a random variable uniformly distributed between -0.5 and +0.5. For n=1 the sample range is just 0.

As the sample size increases the sample maximum approaches a spike function at 0.5 and the sample minimum as a spike function at -0.5. Therefore the distribution of the sample range approaches a spike function at 1.0.

This indicates the necessity that for an extension of the central limit theorem to apply, the sample statistic must be representable as a sum.