  We study of the connection between operator valued central limits for
monotone, Boolean and free probability theory, which we shall call the arcsine,
Bernoulli and semicircle distributions, respectively. In scalar-valued
non-commutative probability these measures are known to satisfy certain
arithmetic relations with respect to Boolean and free convolutions. We show
that generally the corresponding operator-valued distributions satisfy the same
relations only when we consider them in the fully matricial sense introduced by
Voiculescu. In addition, we provide a combinatorial description in terms of
moments of the operator valued arcsine distribution and we show that its
reciprocal Cauchy transform satisfies a version of the Abel equation similar to
the one satisfied in the scalar-valued case.
