  The Central Limit Theorem does not hold for strongly correlated stochastic
variables, as is the case for statistical systems close to criticality.
Recently, the calculation of the probability distribution function (PDF) of the
magnetization mode has been performed with the functional renormalization group
in the case of the three-dimensional Ising model [Balog et al., Phys. Rev.
Lett. {\bf 129}, 210602 (2022)]. It has been shown in that article that there
exists an entire family of universal PDFs parameterized by
$\zeta=\lim_{L,\xi_\infty\rightarrow\infty} L/\xi_\infty$ which is the ratio of
the system size $L$ to the bulk correlation length $\xi_{\infty}$ with both the
thermodynamic limit and the critical limit being taken simultaneously. We show
how these PDFs or, equivalently, the rate functions which are their logarithm,
can be systematically computed perturbatively in the $\epsilon=4-d$ expansion.
We determine the whole family of universal PDFs and show that they are in good
qualitative agreement with Monte Carlo data. Finally, we conjecture on how to
significantly improve the quantitative agreement between the one-loop and the
numerical results.
