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I was grading my Master projects this morning and came upon this graph:

which compares the variability of an importance-sampling estimator versus its self-normalised alternative… This is an interesting case in that self-normalisation does considerably degrade the quality of the approximation in that setting. In other cases, self-normalisation may bring a clear improvement. (This reminded me of a recent email from David Einstein complaining about imprecisions in the importance section of Monte Carlo Statistical methods , incl. the fact that self-normalisation was not truly addressing the infinite variance issue. His criticism is appropriate, we should rewrite this section towards more precise statements…)

Maybe this is to be expected. Here is a similar comparison for finite and infinite variance cases:

The outcome is shown above, with an increased variability in the finite variance case (df=.5, left) and a (meaningful?) decrease in the infinite variance case (df=2.5, right).