Statistical population moments may be finite or infinite. Determining whether certain moments of a population are finite or not based on a finite sample turns out to be a very daunting and difficult task. If one assumes stock returns to behave according the sum stable law, characteristic exponent point estimates of approximately 1.5 are found for Australian stocks. This result is fully in line with previous US findings and implies that the population variance is infinite. Hill-estimates, on the other hand, are above 2 for all stocks, indicating that the second moments do exist. This conflicting result is resolved by showing that the (unconditional) sum stable hypothesis can be rejected firmly. We do this by setting up a simulation experiment, in which we show that combinations of the Hill-estimate and the characteristic exponent produced by the real data are extremely unlikely for sum stables. These results confirm the existence of at least second moments. There is a good chance that third moments exist as well but this calls for further research.
