  Building on Talagrand's proof of the Hoffmann-J{\o}rgensen inequality for
$L_p$ spaces and its version for the exponential Orlicz spaces we provide a
full characterization of Orlicz functions $\Psi$ for which an analogous
inequality holds in the Orlicz space $L_\Psi(F)$, where $F$ is an arbitrary
Banach space.
  As an application we present a characterization of Talagrand-type
concentration inequality for suprema of empirical processes with envelope in
$L_\Psi$ (equivalently for sums of independent $F$-valued random variables in
$L_\Psi(F)$). This result generalizes in particular an inequality by the
first-named author concerning exponentially integrable summands and a recent
inequality due to Chamakh-Gobet-Liu on summands with $\beta$-heavy tails.
Another corollary concerns concentration for convex functions of independent,
unbounded random variables, generalizing recent results due to
Klochkov-Zhivotovskiy and Sambale.
  We also obtain a corollary concerning boundedness in $L_\Psi(F)$ of partial
sums of a series of independent random variables, generalizing the original
result by Hoffmann-J{\o}rgensen.
