Characterizations of B-valued concentration inequalities via the Rademacher type

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued (B-valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type p for some 1 <= p <= 2; which is equivalent to that the Bernstein concentration inequality holds for such B-valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group.