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

## TL;DR

This paper extends strong laws of large numbers to intermediately trimmed Birkhoff sums in certain dynamical systems with infinite mean, matching rates known for independent variables, and applies to systems like interval maps.

## Contribution

It generalizes strong laws of large numbers for intermediately trimmed sums to dependent dynamical systems with infinite mean, including specific conditions and examples.

## Key findings

- Validates the laws for systems with spectral gap property
- Establishes trimming rates comparable to i.i.d. cases
- Applies results to piecewise expanding interval maps

## 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.