 Sharpness of the linear growth bound for integrable observables

For every $0<a<1$, choose $1<p<1/a$. On the Bernoulli shift over independent uniform coordinates in $(0,1)$, the observable $f(x)=x_0^{-1/p}$ is integrable, but $\mu (f(T^{n}x)>n^a)=n^{-ap}$. The events are independent and their probabilities have divergent sum, so the Borel-Cantelli lemma implies that $f(T^{n}x)/n^a>1$ infinitely often almost surely. No exponent below one gives a universal bound for all integrable observables.