  Sturm's strong law of large numbers in $\mathrm{CAT}(0)$ spaces has been an
influential tool to study the geometric mean or also called Karcher barycenter
of positive definite matrices. It provides an easily computable stochastic
approximation based on inductive means. Convergence of a deterministic version
of this approximation has been proved by Holbrook, providing his "nodice"
theorem for the Karcher mean of positive definite matrices. The Karcher mean
has also been extended to the infinite dimensional case of positive operators
on a Hilbert space by Lawson-Lim and then to probability measures with bounded
support by the second author, however the $\mathrm{CAT}(0)$ property of the
space is lost and one defines the mean as the unique solution of a nonlinear
operator equation on a convex Banach-Finsler manifold. The formulations of
Sturm's strong law of large numbers and Holbrook's "nodice" approximation are
natural and both conjectured to converge, however all previous techniques of
their proofs break down, due to the Banach-Finsler nature of the space. In this
paper we prove both conjectures by establishing the most general $L^1$-form of
Sturm's strong law of large numbers and Holbrook's "nodice" theorem in the
operator norm by developing a stochastic discrete-time resolvent flow for the
Karcher barycenter using its Wasserstein contraction property.
