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

Abstract

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necessary conditions for our limit laws. As a side result we obtain strong laws of large numbers for truncated Birkhoff sums.

37 pages

References in corpus (4)

• Pointwise convergence of Birkhoff averages for global observables
• Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean
• Dynamical Borel-Cantelli lemmas and rates of growth of Birkhoff sums of non-integrable observables on chaotic dynamical systems
• On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

Cited by in corpus (4)

• Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type
• Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails
• Trimmed sums for observables on the doubling map
• Almost sure limit theorems with applications to non-regular continued fraction algorithms