An $m-$plet $A$ of positive-definite $N$ by $N$ matrices may be
assigned a Cartan centroid alias Karcher mean alias geometric mean
$G$. The author shows that its usual stochastic interpretation may
be paralleled by a non-stochastic one, explained in main theorem. It
states a certain limiting behavior of compositions of asymmetric
averages (of just two matrices) and leads, {\it i.a.}, to a
facilitated proof of the monotonicity of $G$ with respect to the
Loewner (partial) ordering of $A$s.

MR3027448 Holbrook, John No dice: a deterministic approach to the
Cartan centroid. J. Ramanujan Math. Soc. 27 (2012), no. 4, 509--521.
47A64 (15B48)