Building on Talagrand's proof of the Hoffmann-J{\o}rgensen inequality for
Lp spaces and its version for the exponential Orlicz spaces we provide a
full characterization of Orlicz functions Ψ for which an analogous
inequality holds in the Orlicz space LΨ(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
LΨ (equivalently for sums of independent F-valued random variables in
LΨ(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 β-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 LΨ(F) of partial
sums of a series of independent random variables, generalizing the original
result by Hoffmann-J{\o}rgensen