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.

Abstract. We show that for every sequence of non-negative i.i.d. random variables with infinite mean there exists a proper moderate trimming such that for the trimmed sum process a nontrivial strong law of large numbers holds. We provide an explicit procedure to find a moderate trimming sequence even if the underlying distribution function has a complicated structure, e.g. has no regularly varying tail distribution.

We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St. Petersburg type distribution functions. To prove these statements we introduce the space of quasi-Hölder continuous functions for subshifts of finite type.

On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails. Under a mild mixing condition we show that the appropriately normed and 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 ful lling a spectral gap property and an observable with regularly varying tails but without the assumed mixing condition such that mean convergence fails.

The classic Thue-Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied in the past, including various scaling properties and a partly heuristic multifractal analysis. Some of the difficulties emerge from the appearance of an unbounded potential in the thermodynamic formalism. It is the purpose of this article to review and prove some of the observations that were previously established via numerical or scaling arguments.2010 Mathematics Subject Classification. 37D35, 37C45, 52C23.