  We consider a conservative ergodic measure-preserving transformation $T$ of a
$\sigma$-finite measure space $(X,\mathcal{B},\mu)$ with $\mu(X)=\infty$. Given
an observable $f:X\to \mathbb{R}$ we study the almost sure asymptotic behaviour
of the Birkhoff sums $S_Nf(x) := \sum_{j=1}^N\, (f\circ T^{j-1})(x)$. In
infinite ergodic theory it is well known that the asymptotic behaviour of
$S_Nf(x)$ strongly depends on the point $x\in X$, and if $f\in L^1(X,\mu)$,
then there exists no real valued sequence $(b(N))$ such that $\lim_{N\to\infty}
S_Nf(x)/b(N)=1$ 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 $(\alpha(N))$ and $m\colon X\times
\mathbb{N}\to\mathbb{N}$ such that for $f\in L^1(X,\mu)$ we have
$\lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N)=1$ for $\mu$-a.e. $x\in X$.
Moreover if $f\not\in L^1(X,\mu)$ we give conditions on the induced observable
such that there exists a sequence $(G(N))$ depending on $f$, for which
$\lim_{N\to\infty} S_{N}f(x)/G(N)=1$ holds for $\mu$-a.e. $x\in X$.
