This work proposes block-coordinate fixed point algorithms with applications
to nonlinear analysis and optimization in Hilbert spaces. The asymptotic
analysis relies on a notion of stochastic quasi-Fej\'er monotonicity, which is
thoroughly investigated. The iterative methods under consideration feature
random sweeping rules to select arbitrarily the blocks of variables that are
activated over the course of the iterations and they allow for stochastic
errors in the evaluation of the operators. Algorithms using quasinonexpansive
operators or compositions of averaged nonexpansive operators are constructed,
and weak and strong convergence results are established for the sequences they
generate. As a by-product, novel block-coordinate operator splitting methods
are obtained for solving structured monotone inclusion and convex minimization
problems. In particular, the proposed framework leads to random
block-coordinate versions of the Douglas-Rachford and forward-backward
algorithms and of some of their variants. In the standard case of m=1 block,
our results remain new as they incorporate stochastic perturbations