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, mu) with mu a sigma-finite measure and mu(X) = infinity. 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) : Sigma(N)(j=1) (g circle Tj-1)(x) strongly depends on the point x is an element of X, and that there exists no sequence (d(N)) for which S(N)g(x)/d(N) -> 1 for mu-almost every x is an element of X. In this paper we consider the case g is not an element of L-1(X, mu) 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 S(N)g(x) for an unbounded observable g may be obtained using two methods, addition to S(N)g 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 Renyi 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.