  We introduce and study a new type of convolution of probability measures
called the orthogonal convolution, which is related to the monotone
convolution. Using this convolution, we derive alternating decompositions of
the free additive convolution of compactly supported probability measures in
free probability. These decompositions are directly related to alternating
decompositions of the associated subordination functions. In particular, they
allow us to compute free additive convolutions of compactly supported measures
without using free cumulants or R-transforms. In simple cases, representations
of the corresponding Cauchy transforms as continued fractions are obtained in a
natural way. Moreover, this approach establishes a clear connection between
convolutions and products associated with the main notions of independence
(free, monotone and boolean) in noncommutative probability. Finally, our result
leads to natural decompositions of the free product of rooted graphs.
