Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

We consider a conservative ergodic measure-preserving transformation T of a σ-finite measure space (X,B,μ) with μ(X)=∞. Given an observable f:X→ℝ we study the almost sure asymptotic behaviour of the Birkhoff sums S_Nf(x). In infinite ergodic theory it is well known that the asymptotic behaviour of SNf(x) strongly depends on the point x∈X, and if f∈L^1(X,μ), then there exists no real valued sequence (b(N)) such that lim S_Nf(x)/b(N)=1 almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence (α(N)) and m:X×ℕ→ℕ such that for f∈L^1(X,μ) we have lim S_{N+m(x,N)}f(x)/α(N)=1 for μ-a.e. x∈X. Moreover if f∉L1(X,μ) we give conditions on the induced observable such that there exists a sequence (G(N)) depending on f, for which lim S_Nf(x)/G(N)=1 holds for μ-a.e. x∈X.