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

Abstract

We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,B,\mu )$ with $\mu$ a $\sigma$-finite measure and $\mu \left(X\right)=\infty$. Given an observable $g:X\to R$, it is well known from results by Aaronson, see Aaronson (1997), that in general the asymptotic behaviour of the Birkhoff sums $S_{N}g\left(x\right):={\sum }_{j=1}^N(g\circ T^{j-1})\left(x\right)$ strongly depends on the point $x\in X$, and that there exists no sequence $\left(d_N\right)$ for which $S_{N}g\left(x\right)/d_N\to 1$ for $\mu$-almost every $x\in X$. In this paper we consider the case $g⁄\in L^1(X,\mu )$ 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 $S_{N}g\left(x\right)$ 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 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.