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