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

We consider a conservative ergodic measure-preserving transformation T of the measure space (X,B,μ) with μ a σ-finite measure and μ(X)=∞. Given an observable g:X→R, it is well known from results by Aaronson, see Aaronson (1997), that in general the asymptotic behaviour of the Birkhoff sums SNg(x):=∑j=1N(g∘Tj−1)(x) strongly depends on the point x∈X, and that there exists no sequence (dN) for which SNg(x)/dN→1 for μ-almost every x∈X. In this paper we consider the case g⁄∈L1(X,μ) and continue the investigation initiated in Bonanno and Schindler (2022). We show that for transformations T with strong mixing assumptions for the induced map on a finite measure set, the almost sure asymptotic behaviour of SNg(x) for an unbounded observable g may be obtained using two methods, addition to SNg of a number of summands depending on x and trimming. The obtained sums are then asymptotic to a scalar multiple of N. The results are applied to a couple of non-regular continued fraction algorithms, the backward (or Rényi type) continued fraction and the even-integer continued fraction algorithms, to obtain the almost sure asymptotic behaviour of the sums of the digits of the algorithms.