# Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

## TL;DR

This paper proves mean convergence of trimmed Birkhoff sums for certain observables in dynamical systems with spectral gap, highlighting the importance of mixing conditions and extending results to i.i.d. cases.

## Contribution

It establishes mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails under spectral gap and mild mixing conditions, a novel result even for i.i.d. variables.

## Key findings

- Mean convergence holds under spectral gap and mixing conditions.
- Counterexample shows convergence can fail without mixing.
- Results extend to i.i.d. random variables.

## Abstract

On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails and fulfilling a mild mixing condition. We show that the normed trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable with regularly varying tails but without the assumed mixing condition such that mean convergence fails.