The conduct of the sample average when the first moment is infinite

Many books about probability and statistics only mention the weak and the strong law of large numbers for samples from distributions with finite expectation. However, these laws also hold for distributions with infinite expectation and then the sample average has to go to infinity with increasing sample size. Being curious about the way in which this would happen, we simulated increasing samples (up to n= 40000) from three distributions with infinite expectation. The results were somewhat surprising at first sight, but understandable after some thought. Most statisticians, when asked, seem to expect a gradual increase of the average with the size of the sample. So did we. In general, however, this proves to be wrong and for different parent distributions different types of conduct appear from this experiment. The samples from the "absolute Cauchy"-distribution are most interesting from a practical point of view: the average takes a high jump from time to time and decreases in between. In practice it might well happen, that the observations causing the jumps would be discarded as outlying observations.