  The influence of the tail features of the local magnetic field probability
density function (PDF) on the ferromagnetic Ising model is studied in the limit
of infinite range interactions. Specifically, we assign a quenched random field
whose value is in accordance with a generic distribution that bears platykurtic
and leptokurtic distributions depending on a single parameter $\tau < 3$ to
each site. For $\tau< 5/3$, such distributions, which are basically Student-$t$
and $r$-distribution extended for all plausible real degrees of freedom,
present a finite standard deviation, if not the distribution has got the same
asymptotic power-law behavior as a $\alpha $-stable L\'evy distribution with
$\alpha = (3 - \tau)/(\tau - 1)$. For every value of $\tau $, at specific
temperature and width of the distribution, the system undergoes a continuous
phase transition. Strikingly, we impart the emergence of an inflexion point in
the temperature-PDF width phase diagrams for distributions broader than the
Cauchy-Lorentz ($\tau = 2$) which is accompanied with a divergent free energy
per spin (at zero temperature).
