Almost sure limit theorems with applications to non-regular continued fraction algorithms

We consider an ergodic measure preserving dynamical system (X,T) and an observable f mapping X to the reals. By a theorem by Aaronson there is no strong law of large numbers if either 1) X is a probability measure space and f is non-integrable or if 2) X has infinite measure und f is integrable. While in the situation 1) one can use trimming, i.e. deleting a number of the largest entries, to still obtain a strong law of large numbers, in case of 2) it is possible to add additional summands to obtain a strong law of large numbers. In this talk we will study the situation 2) and also the situation of an infinite measure space and an observable which is non-integrable on a finite measure set. Examples of such a situation are different non-standard continued fraction digits like backwards continued fractions. This is joint work with Claudio Bonanno.