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We consider dynamical systems on a finite measure space fulfilling a spectralgap property and Birkhoff sums of a non-negative, non-integrable observable.For such systems we generalize strong laws of large numbers for intermediatelytrimmed sums only known for independent random variables. The results split upin trimming statements for general distribution functions and for regularlyvarying tail distributions. In both cases the trimming rate can be chosen inthe same or almost the same way as in the i.i.d. case. As an example we showthat piecewise expanding interval maps fulfill the necessary conditions for ourlimit laws. As a side result we obtain strong laws of large numbers fortruncated Birkhoff sums.

Strong laws of large number for intermediately trimmed Birkhoff sums of observables with infinite mean