  In an infinitesimal probability space we consider operators which are
infinitesimally free and one of which is infinitesimal, in that all its moments
vanish. Many previously analysed random matrix models are captured by this
framework. We show that there is a simple way of finding non-commutative
distributions involving infinitesimal operators and apply this to the
commutator and anticommutator. We show the joint infinitesimal distribution of
an operator and an infinitesimal idempotent gives us the Boolean cumulants of
the given operator. We also show that Boolean cumulants can be expressed as
infinitesimal moments thus giving matrix models which exhibit asymptotic
Boolean independence and monotone independence. Finally we demonstrate a
connection to the Markov-Krein transform.
